We provide permutations and combinations practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on permutations and combinations skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.

#### List of permutations and combinations Questions

Question No | Questions | Class |
---|---|---|

1 | Two men enter a railway compartment having 6 seats unoccupied. In how many ways can they are seated? | 11 |

2 | Evaluate the following: ( n+1 C_{n} ) | 11 |

3 | Evaluate ( ^{4} boldsymbol{C}_{3}+^{4} boldsymbol{C}_{4}=? ) | 11 |

4 | The number of permutations of the letters of the word “ENGINEERING” is A ( cdot frac{11 !}{3 ! ! ! !} ) в. ( frac{11 !}{(3 ! 2 !)^{2}} ) c. ( frac{11 !}{(3 !)^{2} cdot 2 !} ) D. ( frac{11 !}{3 !(2 !)^{2}} ) | 11 |

5 | Determine ( n ) if ( ^{2 n} C_{3}:^{n} C_{3}=11: 1 ) | 11 |

6 | How many 3 digit even numbers can be formed using the digits ( 3,5,7,8,9, ) if the digits are not repeated? | 11 |

7 | Find the number of combinations and permutations of 4 letters taken from the word ( boldsymbol{E} boldsymbol{X} boldsymbol{A} boldsymbol{M} boldsymbol{I} boldsymbol{N} boldsymbol{A} boldsymbol{T} boldsymbol{I} boldsymbol{N} ) | 11 |

8 | A car driver knows four different routes from Delhi to Amritsar. From Amritsar to Pathankot, he knows three different routes and from Pathankot to Jammu he knows two different routes. How many routes does he know from Delhi to Jammu? A .4 B. 8 ( c cdot 12 ) ( D cdot 24 ) E. 36 | 11 |

9 | In how many of the permutations of ( n ) things taken ( r ) at a time will 5 things (i) always occur, (ii) never occur? | 11 |

10 | Re. 1 and ( R ) s. 5 coins are available ( ( ) as many required). Find the smallest payment which cannot be made by these coins, if not more than 5 coins are allowed. ( A cdot 3 ) B. 12 ( c cdot 14 ) D. 18 | 11 |

11 | A rectangle polygon of ( n ) sides is constructed. No. of ways 3 vertices be selected so that no two vertices are consecutive is ( mathbf{A} cdot^{n} C_{3}-n(n-4) ) B . ( ^{n} C_{3}-n-n(n-4) ) C. ( ^{n} C_{3}+n-n(n-4) ) D. ( ^{n} C_{3}+n(n-4) ) | 11 |

12 | How many four-letter computer passwords can be formed using only the symmetric letters (no repetition allowed)? A .7920 в. 330 c. 14640 D. 419430 | 11 |

13 | The value of ( ^{10} boldsymbol{C}_{1}+^{10} boldsymbol{C}_{2}+^{10} boldsymbol{C}_{3}+ldots+ ) ( ^{10} mathrm{C}_{9} ) is A ( .2^{10} ) B . ( 2^{11} ) ( c cdot 2^{10}-2 ) D. ( 2^{10}-1 ) | 11 |

14 | How many zeros in ( 100 ! ? ) | 11 |

15 | A college offers 7 courses in the morning and 5 in the evening. Find possible number of choices with the student who want to study one course in the morning and one in the evening. A . 35 B. 12 c. 49 D. 25 | 11 |

16 | ( mathbf{f}^{n} boldsymbol{C}_{boldsymbol{r}-mathbf{1}}=mathbf{3 6},^{boldsymbol{n}} boldsymbol{C}_{boldsymbol{r}}=mathbf{8 4} ) and ( ^{boldsymbol{n}} boldsymbol{C}_{boldsymbol{r}+mathbf{1}}= ) ( 126, ) then ( r ) is ( A cdot 1 ) B . 2 ( c .3 ) D. | 11 |

17 | The value of ( frac{(boldsymbol{n}+mathbf{2}) !-(boldsymbol{n}+mathbf{1}) !}{boldsymbol{n} !} ) is: A ( cdot(n+2) ! ) в. ( (n+1) ! ) c. ( (n+2)^{2} ) D. ( (n+1)^{2} ) ( E ) | 11 |

18 | Given 5 line segments of lengths 2,3,4,5,6 units. Then the number of triangles that can be formed by joining these lines is ( mathbf{A} cdot^{5} C_{3} ) B. ( ^{5} C_{3}-3 ) c. ( ^{5} C_{3}-2 ) D. ( ^{5} C_{3}-1 ) | 11 |

19 | If ( m ) denotes the number of 5 digit numbers if each successive digits are in their descending order of magnitude and ( n ) is the corresponding figure. When the digits and in their ascending order of magnitude then ( (boldsymbol{m}-boldsymbol{n}) ) has the value A ( cdot^{9} C_{4} ) B . ( ^{9} C_{5} ) ( mathrm{c} cdot^{10} mathrm{C}_{3} ) D. ( ^{9} C_{3} ) | 11 |

20 | If ( ^{n} C_{r} ) denotes the number of combinations of ( n ) things taken ( r ) at a time, then the expression ( ^{n} C_{r+1}+^{n} ) ( C_{r-1}+2 times^{n} C_{r} ) equals A. ( ^{n+2} C_{r} ) B. ( ^{n+2} C_{r+1} ) c. ( ^{n+1} C_{r} ) D. ( n+1 C_{r+1} ) | 11 |

21 | There are 7 men and 8 women. In how many ways a committee of 4 members can be made such that a particular woman is always included | 11 |

22 | Find the value of ( ^{10} boldsymbol{C}_{5}+mathbf{2} .left(^{10} boldsymbol{C}_{4}right)+^{10} boldsymbol{C}_{3} ) | 11 |

23 | How many integers with at least one eight and at least one nine as digits are there between 1 and 10000 A. 800 в. 974 ( c .900 ) D. 875 | 11 |

24 | How many 2 – digit numbers can be formed from the digits ( {1,2,3,4, ) 5} without repetition and with repetition? | 11 |

25 | A code word is to consist of two English alphabets followed by two distinct numbers between 1 and ( 9 . ) For example, ( C A 23 ) is a code word. How many such code words are there? | 11 |

26 | If the coefficient of ( (2 r+4)^{t h} ) term and ( (r-2)^{t h} ) term in the expansion of ( (1+ ) ( x)^{18} ) are equal then ( r ) ( mathbf{A} cdot mathbf{9} ) B. 4 ( c .6 ) D. 3 | 11 |

27 | How many integers, greater than 999 but not greater than ( 4000, ) can be formed with the digits 0,1,2,3 and ( 4, ) if repetition of digits is allowed? A . 376 в. 375 ( c .500 ) D. 673 | 11 |

28 | Find the value of (a) ( ^{14} C_{5} ) (b) ( ^{90} C_{2} ) | 11 |

29 | ( mathbf{2} cdot boldsymbol{C}_{mathbf{0}}+mathbf{2}^{2} cdot frac{boldsymbol{C}_{1}}{mathbf{2}}+mathbf{2}^{mathbf{3}} cdot frac{C_{2}}{mathbf{3}}+ldots+ ) ( mathbf{2}^{n+1} cdot frac{C_{n}}{n+1}= ) A ( cdot frac{3^{n+1}-1}{2(n+1)} ) B . ( frac{3^{n+1}-1}{n+1} ) ( mathrm{c} cdot frac{3^{n}-1}{n+1} ) D. ( frac{3^{n+1}}{n+1} ) | 11 |

30 | ( frac{boldsymbol{C}_{mathbf{0}}}{mathbf{1}}+frac{boldsymbol{C}_{mathbf{1}}}{mathbf{2}}+frac{boldsymbol{C}_{mathbf{2}}}{mathbf{3}}+ldots ldots ldots+frac{boldsymbol{C}_{mathbf{1 0 0}}}{mathbf{1 0 1}} ) equals A ( cdot underline{underline{2}^{101}} ) 101 B. ( frac{2^{101}-1}{101} ) c. ( frac{3^{101}}{101} ) D. ( frac{3^{101}-1}{101} ) | 11 |

31 | There are 40 doctors in a surgical department. In how many ways can they be arranged to form the following teams: (a) a surgeon and an assistant; (b) a surgeon and four assistants? | 11 |

32 | Find the number of arrangements of the letters of the word INDEPENDENCE. In how many of these arrangements. Do the vowels never occur together? | 11 |

33 | Find the number of permutations which can be formed out of the letters of the word series taken three together? ( A cdot 32 ) B. 36 c. 42 D. 46 | 11 |

34 | A person always prefers to eat parantha and vegetable dish in his meal. How many ways can he make his plate in a marriage party if there are three types of paranthas, four types of vegetable dishes, three types of salads, and two types of sauces? A . 3360 в. 4096 c. 3000 D. None of these | 11 |

35 | Prove that ( ^{n} C_{0}-^{n} C_{1}+^{n} C_{2}- ) ( ^{n} C_{3}+ldots+(-1)^{r} n_{r}+ldots= ) ( (-1)^{r-1} n-1 C_{r-1} ) | 11 |

36 | ( f^{8} C_{r}=^{8} C_{r+2}, ) then the value of ( ^{r} C_{2} ) is: ( A cdot 8 ) B. 3 ( c cdot 5 ) D. 2 | 11 |

37 | Number greater than 1000 but less than 4000 is formed using the digits 0, 1, 2, 3, 4 (repetition allowed). Their number is [2002] (a) 125 (6) 105 (c) 375 (d) 625 | 11 |

38 | There are ‘ ( m^{prime} ) copies each of ( ^{prime} n^{prime} ) different books in a university library. The number of ways in which one or more than one book can be selected is A ( cdot m^{n}-1 ) B . ( (m+1)^{n}-1 ) C ( cdot(m+1)^{n}-m^{n} ) D. ( (m+1)^{n}-m ) | 11 |

39 | In a game called ‘odd man out’, ( m(m> ) 2) persons toss a coin to determine who will buy refreshment for the entire group. A person who gets an outcome different from that of the rest of the members of the group is called the odd man out. The probability that there is a loser in any game is A. ( 1 / 2 m ) в. ( m / 2^{m-1} ) c. ( 2 / m ) D. none of these | 11 |

40 | Find the unit digit of ( 2 times 3 times 4 times ldots times ) ( mathbf{9 9} ) | 11 |

41 | How many permutations of 4 letters can be made out of the letters of the word examination? | 11 |

42 | How many number greater than a million can be formed with the digits ( 2,3,0,7,7,3,7, ? ) | 11 |

43 | How many three digit numbers can be formed that are divisible by 5 by using the digits 0,2,5,8,9 such that repetition is strictly not allowed A . 21 B . 25 c. 30 D. none of these | 11 |

44 | There are 6 multiple choice questions on an examination. How many sequences of answers are possible, if the first three questions have 4 choices each and the next three have 5 each? | 11 |

45 | In how many ways can the following prizes be given away to a class of 20 students, first and second in Mathematics, first and second in physics, first in Chemistry and first in English? | 11 |

46 | If ( ^{8} C_{r}-^{7} C_{3}=^{7} C_{2}, ) find ( r ) | 11 |

47 | Using the digits 0,2,4,6,8 not more than once in any number, the number of 5 digited numbers that can be formed is A . 16 B. 24 ( c cdot 120 ) D. 96 | 11 |

48 | If ( S_{n}=sum_{r=0}^{n} frac{1}{n} ) and ( t_{n}=sum_{r=0}^{n} frac{r}{n}, ) then ( frac{boldsymbol{t}_{boldsymbol{n}}}{boldsymbol{s}_{boldsymbol{n}}}= ) A ( cdot frac{1}{2} n ) в. ( frac{1}{2} n-1 ) c. ( n-1 ) D. ( frac{2 n-1}{2} ) | 11 |

49 | If ( n-1 C_{6}+^{n-1} C_{7}>^{n} C_{6}, ) then? ( mathbf{A} cdot n>4 ) B ( . n>12 ) ( mathbf{C} cdot n geq 13 ) D. ( n>13 ) | 11 |

50 | The value of ( ^{n} C_{1} ) is | 11 |

51 | A telegraph has 5 arms and each arm is capable of 4 distinct positions, including the position of rest. Find the total number of signals that can be made. | 11 |

52 | The number of permutations of letters of the word “PARALLAL” atken four at a time must be, A .216 в. 244 c. 286 D. 1680 | 11 |

53 | 13. In a shop there are five types of ice-creams available. A child buys six ice-creams. Statement-1: The number of different ways the child can buy the six ice-creams is 10C5. Statement -2 : The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging 6 A’s and 4 B’s in a row. [2008] (a) Statement -1 is false, Statement-2 is true (b) Statement -1 is true, Statement-2 is true; Statement -2 is a correct explanation for Statement-1 (c) Statement -1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1 (d) Statement -1 is true, Statement-2 is false | 11 |

54 | A bag contains Rs. 112 in the form of 1 rupee, 50-paise and 10-paise coins in the ratio ( 3: 8: 10 . ) What is the number of 50 -paise coins? A .112 в. 128 c. 96 D . 24 | 11 |

55 | The value of ( ^{n} C_{n} ) is ( A cdot n ) B. c. 1 D. ( n ! ) | 11 |

56 | In the figure,two 4 -digit numbers are to be formed by filling the places with digits. The number of different ways in which the places can be filled by digits so that the sum of the numbers formed is also a 4 -digit number and in no place the addition is with carrying, is ( mathbf{A} cdot 55^{4} ) B. 220 ( c cdot 45^{4} ) D. none of these | 11 |

57 | Solve the equation ( 3^{x+1} C_{2}+P_{2}^{1} . x= ) ( 4^{x} P_{2}, x in N . ) Find ( x ? ) | 11 |

58 | There are four different white balls and four different black balls. The number of ways that balls can be arranged in a row so that white and black balls are placed alternately is ( A cdot(4 !)^{2} ) B. ( 2(8 !)^{2} ) ( c cdot 4 ) ( D cdot(4 !)^{3} ) | 11 |

59 | 8. The number of arrangements of the letters of the word BANANA in which the two N’s do not appear adjacently is son (2002) (2) 40 (6) 60 (c) 80 (d) 100 | 11 |

60 | Find the number of different 8 letter arrangements that can be made from the letters of the word “DAUGHTER “so that ( (A) ) All vowels occur together ( (B) ) All vowels do not occur together | 11 |

61 | Twenty persons arrive in a town having 3 hotels ( x, y ) and ( z . ) If each person randomly chooses one of these hotels, then what is the probability that aleast 2 of them goes in hotel ( x, ) atleast 1 in hotel ( y ) and atleast 1 in hotel ( z ?( ) each hotel has capacity for more than 20 guests). ( ^{mathbf{A}} cdot frac{^{18} C_{2}}{^{22} C_{2}} ) B. ( frac{20 C_{2} cdot^{18} C_{1} cdot^{17} C_{1} cdot 3^{16}}{3^{20}} ) C. ( frac{^{20} C_{9}}{3^{9}} ) D. ( frac{3^{20}-13.2^{20}+43}{3^{20}} ) | 11 |

62 | The expansion ( ^{n} C_{r}+4 .^{n} C_{r-1}+ ) ( mathbf{6} cdot^{n} boldsymbol{C}_{boldsymbol{r}-mathbf{2}}+mathbf{4} cdot^{boldsymbol{n}} boldsymbol{C}_{boldsymbol{r}-mathbf{3}}+^{boldsymbol{n}} boldsymbol{C}_{boldsymbol{r}-mathbf{4}}= ) A. ( ^{n+4} C_{r} ) В ( cdot 2 cdot^{n+4} C_{r-1} ) ( c cdot 4 cdot^{n} c ) D. ( 11 .^{n} c_{r} ) | 11 |

63 | Number of ways in which 9 different prizes be given to 5 students if one particular boy receives 4 prizes ( & ) the rest of the students can get any numbers of prizes, is: A ( cdot^{9} mathrm{C}_{4} cdot 2^{10} ) B. ( ^{9} mathrm{C}_{5} cdot 5^{4} ) c. ( 4.4^{5} ) D. none | 11 |

64 | The greatest number that can be formed by the digits 7,0,9,8,6,3 A. 9,87,360 В. 9,87,063 c. 9,87,630 D. 9,87,603 | 11 |

65 | ( f^{n} C_{15}=^{n} C_{8}, ) then the value of ( ^{n} C_{21} ) is A .254 в. 250 c. 253 D. None of these | 11 |

66 | If ( 11left[^{n-1} c_{3}right]=24left[^{n} C_{2}right], ) then the value of n is A . 12 B. 1 ( c cdot 10 ) D. 13 | 11 |

67 | How many 10 digits number can be written by using digits (9 and 2) ? ( mathbf{A} cdot^{10} C_{1}+^{9} C_{2} ) B . ( 2^{10} ) ( c cdot^{10} C_{2} ) D. 10 | 11 |

68 | The sides ( A B, B C, C A ) of a triangle ( A B C ) have 3,4 and 5 interior points respectively on them. The number of triangles that can be constructed using these points as vertices is- A . 205 в. 210 ( c .315 ) D. 216 | 11 |

69 | Two series of a question booklet for an aptitude test are to be given to twelve students. In how many ways can the students be placed in two rows of six each so that there should be no identical series side by side and that the students sitting one behind the other should have the same series?? ( mathbf{A} cdot 2 times^{12} C_{6} times(6 !)^{2} ) B. 6! x 6! c. ( 7 ! times 7 ! ) D. ( 2 times 6 ! ) | 11 |

70 | The number of permutations of the letters of the word AGAIN taken three at a time is A .48 B. 24 ( c . ) 36 D. 33 | 11 |

71 | Different calenders for the month of February are made so as to serve for all the coming years. The number of such calenders is ( A cdot 7 ) B. 2 c. 14 D. None of these | 11 |

72 | If ( N ) is the number of positive integral solutions of ( boldsymbol{x}_{1} times boldsymbol{x}_{2} times boldsymbol{x}_{3} times boldsymbol{x}_{4}=mathbf{7 7 0} ) then: This question has multiple correct options A. ( N ) is divisible by 4 district primes B. ( N ) is a perfect square C. ( N ) is a perfect fourth power D. ( N ) is a perfect ( 8^{t h} ) power | 11 |

73 | How many 3 digit numbers can be formed from the digits 2,3,5,6,7,9 which are divisible by 5 and none of the digits is repeated? A . 5 B. 10 c. 15 D. 20 | 11 |

74 | The value of ( ^{95} C_{4}+sum_{j=1}^{5} 100-j C_{3} ) is ( mathbf{A} .^{95} C_{5} ) В. ( ^{100} C_{4} ) ( mathrm{c} .^{99} mathrm{C}_{4} ) D. ( ^{10} C_{5} ) | 11 |

75 | The students in a class are seated according to their marks in the previous examination. Once, it so happens that four of the students got equal marks and therefore the same rank. To decide their seating arrangement, the teacher wants to write down all possible arrangements one in each of separate bits of paper in order to choose one of these by lots. How many bits of paper are required? | 11 |

76 | Total number of 6 -digit number in which only and all the five digits 1,3,5,7 and 9 appear, is : A ( cdot frac{1}{2}(6 !) ) B. ( 5^{text {f }} ) c. ( frac{5}{2}(6 !) ) D. ( 6 ! ) | 11 |

77 | POWIU POSILICI 11. Prove that (1989 – 5 Marks) Co – 22 G + 32 C2 – ……. + (-1)” (n + 1)2 Cn =0, n>2, where C, = “Cr. | 11 |

78 | In how many ways can a football team of 11 players be selected from 16 players? How many of these will Include 2 particular players? ( mathbf{A} cdot 4638,2020 ) B. 4658,2001 c. 4368,2002 D. None of these | 11 |

79 | There are 4 letter boxes in a post office. In how many ways can a man post 8 distinct letters? ( mathbf{A} cdot 4 times 8 ) B . ( 8^{4} ) ( c cdot 4^{8} ) D ( cdot P(8,4) ) | 11 |

80 | How many different four digit numbers can be formed using the digits 1,2,3,4,5,6 when repetition is not allowed and each number starts with ( 1 ? ) A . 60 в. 120 c. 140 D. 25 | 11 |

81 | ( ^{404} boldsymbol{C}_{4}-^{4} boldsymbol{C}_{1}^{303} boldsymbol{C}_{4}+^{4} boldsymbol{C}_{2}^{202} boldsymbol{C}_{4}- ) ( ^{4} C_{3}^{101} C_{4} ) is equal to ( mathbf{A} cdot(401)^{4} ) B . ( (101)^{4} ) c. 0 D. ( (201)^{4} ) | 11 |

82 | Each of the 11 letters ( A, H, I, M, O, T, U, V, W, X ) and ( Z ) appears same when looked at in a mirror. they are called symmetric letters. other letters in the alphabet are asymmetric letters. How many three letter computer passwords can be formed (no repetition allowed) with at least one symmetric letter? | 11 |

83 | How many numbers greater than a million can be formed with the digits ( mathbf{2}, mathbf{3}, mathbf{0}, mathbf{3}, mathbf{4}, mathbf{2}, mathbf{3} ? ) | 11 |

84 | 17. Statement-1: The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is °C; Statement-2: The number of ways of choosing any 3 places from 9 different places is ‘CZ. [2011] (a) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1. (b) Statement-1 is true, Statement-2 is false. © Statement-1 is false, Statement-2 is true. (d) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1. | 11 |

85 | A cricket 11 is to be chosen from 16 players of whom 7 are bowlers, 4 can do the wicketkeeping. Number of ways this can be done to contain exactly 5 bowlers, 2 wicket keepers is A . 945 в. 885 c. 630 D. 715 | 11 |

86 | 24. For any positive integer m, n (with n m ), let Cm n+1 Prove that (-)-(-)-(02)–~-= + + m m / m+2) Prove that Hence or otherwise, prove that n-1 (n-2 +31 + ….. + (n – m +1 mm) ” m) [n+2) m)- m+2). 2000 6 Mar | 11 |

87 | If ( n=^{m} C_{2}, ) then the value of ( ^{n} C_{2} ) is given by A. ( ^{m+1} C_{4} ) в. ( ^{m-1} C_{4} ) c. ( ^{m+2} C_{4} ) D. ( 3 .^{m+1} C_{4} ) | 11 |

88 | Robert was asked to made a 5 digit number from the digits 2 and 4 such that first digit cannot be ( 4 . ) Find the number of such 5 digit numbers that can be formed. A . 10 в. 12 c. 16 D. 18 E. None of these | 11 |

89 | There are 15 buses running between two towns. In how many ways can a man go to one town and return by a different bus? | 11 |

90 | A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected is the team has no girl? | 11 |

91 | Find the ( 4^{t h} ) term of ( left(9 x-frac{1}{3 sqrt{x}}right)^{18} ) A . 16500 B . 18564 c. 16540 D. 32600 | 11 |

92 | A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw? (1986 – 24/, Marks) | 11 |

93 | An automobile dealer provides motor cycles and scooters in three body patterns and 4 different colours each The number of choices open to a customer is ( A cdot^{5} C_{3} ) в. ( ^{4} C_{3} ) c. ( 4 times 3 ) D. ( 4 times 3 times 2 ) | 11 |

94 | A man has 7 trousers and 10 shirts How many different outfits can he wear? | 11 |

95 | The value of the expression ( mathbf{2}^{k}left(begin{array}{l}n \ mathbf{0}end{array}right)left(begin{array}{l}boldsymbol{n} \ boldsymbol{k}end{array}right)-mathbf{2}^{k-1}left(begin{array}{l}boldsymbol{n} \ mathbf{1}end{array}right)left(begin{array}{l}boldsymbol{n}-mathbf{1} \ boldsymbol{k}-mathbf{1}end{array}right)+ ) ( 2^{k-2}left(begin{array}{l}n \ 2end{array}right)left(begin{array}{l}n-2 \ k-2end{array}right) dots+ ) ( (-1)^{k}left(begin{array}{c}n \ kend{array}right)left(begin{array}{c}n-k \ 0end{array}right) ) is ( A cdotleft(begin{array}{l}n \ kend{array}right. ) ( ^{mathbf{B}} cdotleft(begin{array}{c}n+1 \ kend{array}right) ) c. ( left(begin{array}{c}n+1 \ k+1end{array}right) ) D. ( left(begin{array}{l}n-1 \ k-1end{array}right) ) | 11 |

96 | In a family there are 10 adults and 6 children. In how many ways can they be seated around table so that all the children do not sit together? A . 16 B. ( 15 ! ) c. ( 16 !-(11 ! times 6 !) ) D. ( 15 !-(10 ! times 6 !) ) | 11 |

97 | Verify that: ( ^{10} C_{4}+^{10} C_{3}=^{11} C_{4} ) | 11 |

98 | Number of permutations of the word PANCHKULA where ( A & U ) are separated (Note that PAANCHKUL is one such word) is equal to: A ( cdot 15 mid 7 ) B. 21] c. 241 D. ( 36 mid 7 ) | 11 |

99 | A man wears socks of two colours – Black and brown. He has altogether 20 black socks and 20 brown socks in a drawer. Supposing he has to take out of the socks in the dark, how many must he take out to be sure that he has a matching pair? A . 3 B . 20 ( c .39 ) D. None of these | 11 |

100 | In how many ways can four people, each throwing a dice once, make a sum of ( 6 ? ) ( mathbf{A} cdot^{9} C_{2} ) в. ( ^{10} C_{3} ) ( mathbf{c} cdot^{8} C_{3} ) D. ( ^{9} C_{3} ) | 11 |

101 | How many ways are there to arrange the letters in the word GARDEN with vowels in alphabetical order [2004] (a) 480 (6) 240 (c) 360 (d) 120 | 11 |

102 | ( (2 x+3 y)^{5} ) | 11 |

103 | A regular polygon has 20 sides. How many triangles can be drawn by using the vertices but not using the sides? | 11 |

104 | The number of Four digit numbers formed by using the digits 0,2,4,5 and which are not divisible by ( 5, ) is A . 10 B. 8 ( c cdot 6 ) D. 4 | 11 |

105 | In how many ways can 3 people be seated in a row containing 7 seats? | 11 |

106 | Find the number of ways of choosing two squares which are not adjacent in a ( 8 X 8 ) chess board A. 1904 ways B. 904ways c. 1004 ways D. 2904ways | 11 |

107 | The sum ( ^{20} C_{0}+^{20} C_{1}+^{20} C_{2}+ldots . .+^{20} ) ( C_{10} ) is equal to A ( cdot 2^{20}+frac{20 !}{(10 !)^{2}} ) В. ( 2^{19}+frac{1}{2} cdot frac{20 !}{(10 !)^{2}} ) C ( cdot 2^{19}+^{20} C_{10} ) D. none of these | 11 |

108 | There are 8 types of pant pieces and 9 types of shirt pieces with a man. The number of ways in which a pair (1 pant, 1 shirt) can be stitched by the tailor is A . 17 B. 56 c. 64 D. 72 | 11 |

109 | In how many ways can ( n ) things be given to ( boldsymbol{p} ) persons, when there is no restriction as to the number of things each may receive? | 11 |

110 | Anagrams are made by using the letters of the word ( ^{prime} boldsymbol{H} boldsymbol{I} boldsymbol{N} boldsymbol{D} boldsymbol{U} boldsymbol{S} boldsymbol{T} boldsymbol{A} boldsymbol{N}^{prime} ) In how many of these anagrams do the vowels and consonants occupy the same relative positions as in IHINDUST ( boldsymbol{A} boldsymbol{N}^{prime} ) | 11 |

111 | What is the probability of selecting two spade cards from a pack of 52 cards? A . ( 1 / 17 ) B. 15/17 c. ( 32 / 64 ) D. 5/17 | 11 |

112 | The students in a class are seated according to their marks in the previous examination.0nce, it so happens that four of the students got equal marks and therefore the same rank To decide their seating arrangement, the teacher wants to write down all possible arrangements one in each of separate bits of paper in order to choose one of these by lots. How many bits of paper are required? | 11 |

113 | ( f^{20} P_{r}=13 times^{20} P_{r-1}, ) then the value of is | 11 |

114 | A number lock on a suitcase has 3 wheels each labeled with 10 digits from 0 to ( 9 . ) If the opening of the lock is a particular sequence of three digits with no repeats, how many such sequences will be possible? | 11 |

115 | f ( ^{24} C_{x}=^{24} C_{2 x+3}, ) find ( x ) | 11 |

116 | The sum of the series ( ^{20} C_{0}+^{20} C_{1}+ ) ( ^{20} C_{2}+ldots+^{20} C_{10} ) is A ( 2^{20} ) 0 B . ( 2^{19} ) C ( cdot 2^{19}+frac{1}{2} cdot 20 C_{10} ) D – ( 2^{19}-frac{1}{2} cdot 2^{0} C_{10} ) | 11 |

117 | There are 6 books of Economics, 3 of Mathematics and 2 of Accountancy. In how many ways can these be placed on a shelf, if: 1. Books on the same subject are together? 2. Books on the same subject are not together? | 11 |

118 | How many different signals can be made by 5 flags from 8 flags of different colours? | 11 |

119 | Find the number of permutations that can be made with the letters of the word PRAR’? A . ( 4 ! ) в. ( frac{4 !}{2 !} ) c. ( 2 ! times 2 ! ) D. none of these | 11 |

120 | Find the total number of distinct vehicle numbers that can be formed using two letters followed by two numbers. Letters need to be distinct. A . 60000 B. 65000 c. 70000 D. 75000 | 11 |

121 | In how many ways 5 different balls can be distributed into 3 boxes so that no box remains empty? | 11 |

122 | If the number of function from the set ( A={0,1,2} ) and ( B= ) {0,1,2,3,4,5,6,7} such that ( f(i), leq ) ( boldsymbol{f}(boldsymbol{j}), boldsymbol{i}<boldsymbol{j} ; boldsymbol{i}, boldsymbol{j} epsilon boldsymbol{A} mathbf{2 0} boldsymbol{k}, ) then the value of ( k ) is | 11 |

123 | The value of ( sum_{r=1}^{n} frac{n P_{r}}{r !} ) ( mathbf{A} cdot 2^{n} ) B . ( 2^{n}-1 ) ( c cdot 2^{n-1} ) D. ( 2^{n+1} ) | 11 |

124 | The 9 horizontal and 9 vertical lines on an ( 8 times 8 ) chess-board form ( r ) rectangles and ( s ) squares. Then, the ratio ( s: r ) in its lowest terms is A ( cdot frac{1}{6} ) в. ( frac{17}{108} ) c. ( frac{4}{27} ) D. None of the above | 11 |

125 | The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least on ball is | 11 |

126 | The number of ways in which 6 Boys and 5 Girls can sit in a row so that all the girls may be together is A . ( 6 ! times 5 ! ) B . ( 6 !^{7} P_{5} ) c. ( (6 !)^{2} ) D. ( 7 ! times 5 ! ) | 11 |

127 | There are ( n ) points in a plane out of these points no three are in the same straight line except ( p ) points which are collinear. Let ” ( k ) ” be the number of straight lines and “m” be the number of triangles.Then find ( m-k ) ?(Assume ( boldsymbol{n}=boldsymbol{7} boldsymbol{p}=mathbf{5}) ) | 11 |

128 | Assertion In the shop there are five types of ice- creams available, a child buys six ice- creams. Reason The number of different ways the child can buy the six ice-creams is ( ^{10} C_{3} ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect | 11 |

129 | The coefficient of y in the expansion of ( left(y^{2}+frac{c}{y}right)^{5} ) is A . 20c B. 10c ( c cdot 10 c^{3} ) D. ( 20 c^{2} ) | 11 |

130 | 12. The set S = {1, 2, 3, ……., 12} is to be partitioned into three sets A, B, C of equal size. Thus A UBU C = S, AnB = BC = An C = 0. The number of ways to partition Sis [2007] (a) un (6) can4 o 12! o 12! o 12! 12! © 3143 ) 314414 (41)3 | 11 |

131 | Based on this information answer the questions given below. A string of three English letters is formed as per the following rules: (a) The first letter is any vowel. (b) The second letter is ( boldsymbol{m}, boldsymbol{n} ) or ( boldsymbol{p} ) (c) If the second letter is ( m ) then the third letter is any vowel which is different from the first letter. (d) If the second letter is ( n ) then the third letter is ( e ) or ( u ) (e) If the second letter is ( p ) then the third letter is the same as the first letter. How many strings of letters can possibly be formed using the above rules? ( mathbf{A} cdot 40 ) B . 45 c. 30 D. 35 | 11 |

132 | A flag is to be coloured in four stripes by using 6 different colours,no two consecutive stripes being of the same color.This can be done A. 1500 ways B. 750 ways ( c cdot 6^{4} ) ways D. none of these | 11 |

133 | We are required to form different words with the help of the letters of the word INTEGER. Let m1 be the number of words in which I and ( mathrm{N} ) are never together and ( mathrm{m} 2 ) be the number of words which begin with I and end with ( mathrm{R} ), then ( mathrm{m} 1 / mathrm{m} ) 2 is given by: A .42 B. 30 ( c cdot 6 ) D. 1/30 | 11 |

134 | The number of ways in which 20 different white balls and 19 different black balls be arranged in a row. So that no two balls of the same colour come together is A ( cdot 20 !^{2} 1 P_{19} ) B . ( 20 ! times 19 ! ) c. ( (20 !)^{2} ) D. ( (21) !^{120} C_{19} ) | 11 |

135 | How many permutations can be made with the letters of the word ‘MEADOW ( S^{prime} ) such that the vowels occupy even places? A. 720 в. 144 ( c cdot 120 ) D. 36 | 11 |

136 | From a well shuffled pack of 52 playing cards two cards drawn at random. The probability that either both are red or both are kings is: ( ^{mathrm{A}} cdot frac{left(^{26} C_{2}+^{4} C_{2}right)}{^{52} C_{2}} ) B. ( frac{left(^{26} C_{2}+^{4} C_{2}-^{2} C_{2}right)}{^{52} C_{2}} ) ( ^{mathrm{c}} cdot frac{30}{^{52} C_{2}} ) D. ( frac{39}{52} C_{2} ) | 11 |

137 | How many numbers can be formed by using all the digits 1,2,3,4,3,2,1 so that the odd digits always occupy the odd places? A . 16 B. 17 c. 18 D. 20 | 11 |

138 | If ( n=m_{2}, ) then the value of ( ^{n} C_{2} ) is given by ( mathbf{A} cdot m+1 C_{4} ) В . ( m-1 C_{4} ) c. ( m+2 C_{4} ) D. None of these | 11 |

139 | There are 10 railway stations between a station ( x ) and another station ( y ) Find the number of different tickets that must be printed so as to enable a passenger to travel from any one station to any other | 11 |

140 | Find the numerically greatest term in the expansion ( (3-5 x)^{15} ) when ( x=frac{1}{5} ) | 11 |

141 | Consider the letters of the word ‘MATHEMATICS’. Possible number of words in which no two vowels are together is A ( cdot 7 !^{8} C_{2} frac{4 !}{2 !} ) в. ( frac{7 !^{8}}{2 !} C_{4} frac{4 !}{2 !} ) c. ( frac{7 !^{8}}{2 ! 2 !} C_{4} frac{4 !}{2 !} ) D. ( frac{7 !}{2 ! 2 ! 2 !}^{8} C_{4} frac{4 !}{2 !} ) | 11 |

142 | UW. 6. Eighteen guests have to be seated, half on each side of a long table. Four particular guests desire to sit on one particular side and three others on the other side. Determine the number of ways in which the sitting arrangements can be made. (1991 – 4 Marks) | 11 |

143 | Find the number of ways of arranging six persons (having ( A, B, C ) and ( D ) among them) in a row so that ( A, B, C, ) and ( D ) are always in order ABCD (not necessarily together) | 11 |

144 | f 11. The value of 5° ca + § 56-5C is (a) SSC (b) 55. () 56C; [2005 (d) 56cm r= 1 | 11 |

145 | The product of ( r ) consecutive integers is divisible by This question has multiple correct options A. ( ^{text {В }} sum_{k=1}^{r-1} k ) ( c cdot r ) D. none of these | 11 |

146 | The number of six-digit numbers which have sum of their digits as an odd integer, is A . 45000 B. 450000 c. 97000 D. 970000 | 11 |

147 | lului al lulul N. If(1 + x)” = Co + C x + C x2 + …… +Cxthen show that the sum of the products of the C’s taken two at a time, represented by E Ecc; is equal to 22n-1 _ (2n)! 0<i<j<n 2(n!) | 11 |

148 | Find the distinct permutations of the letters of the word MISSISSIPPI? | 11 |

149 | Find the number of four letter words, with meaning or without meaning, that can be formed by using the letters of the word’ ( C H E M I S T R Y^{prime} ) A . 1024 B. 3251 ( c .3024 ) D. 24 | 11 |

150 | A password for a computer system requires exactly 6 characters. Each character can be either one of the 26 letters from ( A ) to ( Z ) or one of the ten digits from 0 to ( 9 . ) The first character must be a letter and the last character must be a digit. How many different possible passwords are there? A ( cdot ) Less than ( 10^{7} ) B. Between ( 10^{7} ) and ( 10^{8} ) C . Between ( 10^{8} ) and ( 10^{9} ) D. Between ( 10^{9} ) and ( 10^{10} ) E. More than ( 10^{10} ) | 11 |

151 | 20 persons were invited to a party. In how many ways can they be seated in a round table such that two particular persons sit on either side of the host? A . 18 в. ( 16 ! ) c. ( 2 times 18 ) D. none of these | 11 |

152 | 13. The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is (2012) (a) 75 (b) 150 (c) 210 (d) 243 | 11 |

153 | Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First, the women choose the chairs amongst the chairs marked 1 to 4 then the men select the chairs amongst the remaining. The number of possible arrangements are: | 11 |

154 | When six fair coins are tossed simultaneously, in how many of the outcomes will at most three of the coins turn up as heads? A . 25 B. 41 ( c cdot 22 ) D. 42 | 11 |

155 | Out of 7 consonants and 4 vowels, words are formed each having 3 consonants and 2 vowels. The number of such words that can be formed is A . 210 B. 25200 c. 2520 D. 302400 | 11 |

156 | The letters of the word ( C ) OC ( H I N ) are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is | 11 |

157 | The number of ways of arranging ( boldsymbol{m} ) positive and ( n(<m+1) ) negative signs in a row so that no two negative signs are together is A ( .^{m+1} P_{n} ) B. ( n+1 P_{m} ) ( mathrm{c} cdot^{m+1} C_{n} ) D. ( n+1 C_{m} ) | 11 |

158 | At an election meeting 10 speakers are to address the meeting. The only protocol to be observed is that whenever they speak P.M. will speak before M.P. and M.P. will speak before M.L.A. In how many ways can the meeting be addressed? | 11 |

159 | The different six digit numbers whose 3 digits are even and 3 digits are odd is A . 281250 B. 281200 ( c cdot ) 156250 D. none of these | 11 |

160 | ( f^{n} P_{r}=840,^{n} C_{r}=35, ) then ( n ) is equal to: ( A cdot 6 ) B. ( c cdot 8 ) D. | 11 |

161 | In an eleventh storey building 9 people enter a lift cabin. It is known that they will leave the lift in group of 2,3 and 4 at different residential storey. Find the number of ways in which they can get down | 11 |

162 | Four couples (husband and wife) decido to form a committee of four members. The number of different committees that can be formed in which no couple finds a place Is | 11 |

163 | How many words can be formed with the letters of the word ‘OMEGA’ when vowels being never together? A . 12 B. 36 ( c cdot 24 ) D. 84 | 11 |

164 | If ( ^{49} boldsymbol{C}_{3 r-2}=^{49} boldsymbol{C}_{2 r+1} ) find ( ^{prime} boldsymbol{r}^{prime} ) | 11 |

165 | 24. · From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is : [JEEM 20181 (a) less than 500 (b) at least 500 but less than 750 C at least 750 but less than 1000 (d) at least 1000 | 11 |

166 | There are five boys ( A, B, C, D, E . ) The order of their height is ( A<B<C<D<E ) Number of ways in which they have to be arranged in 4 seats in increasing order of their height such that ( C & E ) are never adjacent, is | 11 |

167 | golfn-Ic=(K²3) “Erti thenke la -2, -1/5) [ e c I-V3 31 (4827 | 11 |

168 | The letters of word ( O U G H T ) are written in all possible orders and these words are written out as in a dictionary. Find the rank of the word ( T O U G H ) in this dictionary. ( A cdot 89 ) B. 90 c. 91 D. 92 | 11 |

169 | ( sum_{r=0}^{m} n+r C_{n}= ) A. ( ^{n+m+1} C_{n+1} ) В. ( ^{n+m+2} C_{n} ) ( mathbf{c} cdot n+m+3 C_{n+1} ) D. none of these | 11 |

170 | How many numbers amongst the numbers 9 to 54 are there which are exactly divisible by 9 but not by ( 3 ? ) ( A cdot 8 ) B. 6 ( c cdot 5 ) D. Nil | 11 |

171 | 19. Assuming the balls to be identical except for difference in colours, the number of ways in which one or more balls can be selected from 10 white, 9 green and 7 black balls is : [2012] (a) 880 (6) 629 (c) 630 (d) 879 | 11 |

172 | There are three copies each of four different books. The number of ways in which they can be arranged in a shelf is A ( cdot frac{12 !}{(3 !)^{4}} ) в. ( frac{12 !}{(4 !)^{3}} ) c. ( frac{12 !}{(3 !)^{4} 4 !} ) D. ( frac{12 !}{(4 !)^{3} 3 !} ) | 11 |

173 | Seven different coins are to be divided amongst three persons. If no two of the persons receive the same number of coins but each receives at least one coin ( & ) none is left over, then the number of ways in which the division may be made is | 11 |

174 | f ( ^{n} C_{4},^{n} C_{5} ) and ( ^{n} C_{6} ) are in AP, then ( n ) is A . 7 or 14 B. 7 c. 14 D. 14 or 21 | 11 |

175 | The coefficient of ( x^{18} ) in the expansion of ( (1+x)(1-x)^{10}left{left(1+x+x^{2}right)^{9}right} ) is? ( mathbf{A} cdot 84 ) в. 126 ( c cdot-42 ) D. 42 | 11 |

176 | If ( ^{28} C_{2 r}:^{24} C_{2 r-4}=225: 11, ) find ( r ) | 11 |

177 | The value of ( ^{6} C_{4} ) is ( mathbf{A} cdot mathbf{6} ) B. 9 c. 15 D. 240 | 11 |

178 | The sum of the digits in the unit place of all the numbers formed with the help of 3,4,5,6 taken all at a time, is A . 432 в. 108 ( c .36 ) D. 18 | 11 |

179 | 26. A committee of 11 members is to be formed from 8 males and 5 females. Ifm is the number ofways the committee is formed with at least 6 males and n is the number of ways the committee is formed with at least 3 females, then: [JEEM 2019-9 April (M) (a) m+n=68 (b) m=n=78 (C) n=m-8 (d) m=n=68 | 11 |

180 | if ( ^{2017} c_{0}+^{2017} c_{1}+^{2017} c_{2}+ ) ( ldots+^{2017} c_{1008}=lambda^{2}(lambda>0), ) then remainder when ( lambda ) is divided by 33 is ( A cdot 8 ) B. 13 ( c cdot 17 ) D. 25 | 11 |

181 | The value of ( ^{n-1} C_{r-2}+^{n-1} C_{r-1} ) is A. ( ^{n-2} C_{r} ) B. ( ^{n-1} C_{r} ) c. ( ^{n+1} C_{r} ) D. ( ^{n} C_{r-1} ) | 11 |

182 | In how many ways is it possible to choose a white square and a black square on a chess board so that the squares must not lie in the same row or column – A . 56 в. 896 c. 60 D. 768 | 11 |

183 | Dondue. A committee of 12 is to be formed from 9 women and men, ow many ways this can be done if at least five women have to be included in a committee? In how many of these committees (1994 – 4 Marks) (a) The women are in majority? (b) The men are in majority? | 11 |

184 | Find the number of words each consisting of 3 vowels and 3 consonants that can be formed from the letters of the word Circumference. A . 22100 B . 40020 c. 32400 D. 64000 | 11 |

185 | There are 6 items in column-A and 6 items in column-B. A student is asked to match each item in column-A with an item in column-B. The number of possible (correct or incorrect) answers are there to this question is A . 720 B. 620 c. 520 D. 820 | 11 |

186 | The number of ways, in which 12 identical coins can be put into 5 different purses, if none of the purses remain empty, is A. 660 B. 110 ( c cdot 165 ) D. 330 | 11 |

187 | ( sum_{k=1}^{10} k . k != ) A . 10 B. 111 ( c cdot 10 !+1 ) D. 11!-1 | 11 |

188 | Determine ( n ) if ( ^{2 n} C_{2}:^{n} C_{2}=12: 1 ) | 11 |

189 | Assertion If ( ^{n} C_{r-1}=36,^{n} C_{r}=84 ) and ( ^{n} C_{r+1}= ) ( 126, ) then ( ^{r} C_{2}=8 ) Reason ( n C_{r}=^{n} C_{n-r} ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Assertion is incorrect and Reason is correct | 11 |

190 | In how many ways is it possible to choose a white square and a black square on a chessboards, so that the squares must not lie in the same row or column? A . 56 в. 896 c. 60 D. 768 | 11 |

191 | 21. An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at random without replacement from the urn. The probability that the three balls have different colours is [2010] (b) zi ما نیا w ith | 11 |

192 | 22. If all the words (with or without meaning) having five letters, formed using the letters of the word SMALL and arranged as in a dictionary, then the position of the word SMALL IS: [JEE M 2016] (a) 52nd (C) 46th (b) 58th (d) 59th | 11 |

193 | How many alphabets need to be there in a language if one were to make 1 million distinct 3 digit initials using the alphabets of the language? A . 10 в. 100 c. 56 D. 26 | 11 |

194 | The number of intersection points of diagonals of 2009 sides polygon, which lie inside the polygon. A ( .^{2009} mathrm{C}_{4} ) В. ( 2009 mathrm{C}_{2} ) c. ( 2008 C_{4} ) D. ( 2008 mathrm{C}_{2} ) | 11 |

195 | Prove that at any time, the total number of persons on the earth who shake hands an odd number of times is even | 11 |

196 | Choose the correct option for the following. ( boldsymbol{n} !=boldsymbol{n}(boldsymbol{n}-1)(boldsymbol{n}-boldsymbol{2}) dots . . .3 .2 .1 ) A. True B. False c. Ambiguous D. Data insufficient | 11 |

197 | Three horses ( H_{1}, H_{2}, H_{3} ) entered a field which has seven portions marked ( boldsymbol{P}_{1}, boldsymbol{P}_{2}, boldsymbol{P}_{3}, boldsymbol{P}_{4}, boldsymbol{P}_{5}, boldsymbol{P}_{6} ) and ( boldsymbol{P}_{7} . ) If no two horses are allowed to enter the same portion of the field, in-how many ways can the horses graze the grass of the field? | 11 |

198 | The number of combination of 16 things, 8 of which are alike and the rest different, taken 8 at a time is ( ? ) | 11 |

199 | If ( frac{1}{5} C_{r}+frac{1}{6} C_{r}=frac{1}{4 C_{r}}, ) then the value of ( r ) equals to A . 4 B . 2 c. 5 D. 3 | 11 |

200 | Find number of negative integral solution of equation ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=-mathbf{1 2} ) | 11 |

201 | The number of factors (excluding 1 and the expression itself) of the product of ( boldsymbol{a}^{7} boldsymbol{b}^{4} boldsymbol{c}^{3} ) def where ( boldsymbol{a}, boldsymbol{b}, boldsymbol{c}, boldsymbol{d}, boldsymbol{e}, boldsymbol{f} ) are all prime numbers is A. 1278 B. 1360 c. 1100 D. 1005 | 11 |

202 | Find the number of parallelogram formed if 10 parallel lines in a plane are intersected by a family of 12 parallel lines. | 11 |

203 | (d) all 25. Consider a class of 5 girls and 7 boys. The number of different teams consisting of 2 girls and 3 boys that can be formed from this class, if there are two specific boys’ A and B, who refuse to be the members of the same team, is: [JEE M 2019-9 Jan (M) (a) 500 (b) 200 (C) 300 (d) 350 | 11 |

204 | How Many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of the word ( D A U G H T E R ? ) | 11 |

205 | If ( ^{n} C_{10}=^{n} C_{15}, ) find ( ^{27} C_{n} ) | 11 |

206 | Prove that ( frac{^{n} C_{r}}{n-1}=frac{n}{r} ) where ( 1 leq ) ( boldsymbol{r} leq boldsymbol{n} ) | 11 |

207 | If all the words, with or without meaning, are written using the letters of the word QUEEN and are arranged as in English dictionary, then the position of the word QUEEN is. ( mathbf{A} cdot 45^{t h} ) B. ( 46^{text {th }} ) ( mathbf{c} cdot 44^{t h} ) D. ( 47^{t h} ) | 11 |

208 | Evaluate the following: ( ^{35} boldsymbol{C}_{35} ) | 11 |

209 | 7 boys and 8 girls have to sit in a row on 15 chairs numbered from 1 to 15 then? A. Number of ways boys and girls sit alternately is ( 8 ! 7 ! ) B. Number of ways boys and girls sit alternately is ( 2(8 ! 7 !) ) C. The number of ways in which first and fifteenth chair are occupied by boys and between any two boys an even number of girls sit is ( ^{9} C_{4} 8 ! 7 ! ) D. The number of ways in which first and last seat are occupied by boys and between any two boys an even number of girls sit is ( left(2^{9} C_{4} 8 ! 7 !right) ) | 11 |

210 | The value of ( ^{19} C_{18}+^{19} C_{17} ) ( mathbf{A} cdot 1200 ) B. 2000 ( mathbf{C} cdot 190 ) D. None of these | 11 |

211 | How many 3 -digit number can be formed from the digit 1,2,3,4 and 5 assuming that (i) repetition of the digit is allowed? (ii) repetition of the digits is not allowed? | 11 |

212 | Find the sum of all ( 4- ) digit numbers that can be formed using the digits 0,2,4,7,8 without repetition. | 11 |

213 | Ten different letter of an alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have at least one letter repeated is: A . 19670 B. 39758 c. 69760 D. 99748 | 11 |

214 | If ( ^{n} C_{6}:^{n-3} C_{3}=33: 4, ) find ( n ) | 11 |

215 | Consider the following statements: 1. If 18 men can earn Rs. 1,440 in 5 days, then 10 men can earn Rs. 1,280 in 6 days. 2. If 16 men can earn Rs. 1,120 in 7 days, then 21 men can earn Rs. 800 in 4 days. Which of the above statements is/are correct? A. 1 only B. 2 only c. Both 1 and 2 D. Neither 1 nor 2 | 11 |

216 | The number of different ways of distributing 10 marks among 3 questions, each carrying at least 1 mark, is A . 72 B. 71 ( c .36 ) D. none of these | 11 |

217 | If each permutation of the digits ( mathbf{1}, mathbf{2}, mathbf{3}, mathbf{4}, mathbf{5}, mathbf{6} ) are listed in the increasing order of magnitude, then ( 289^{t h} ) term will be A . 361452 B. 321546 c. 321456 D. 341256 | 11 |

218 | Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls ( & 5 ) blue balls. If each selection consists of 3 balls of each color. | 11 |

219 | How many words can be formed with the letters of FAILURE when all the vowels should come together? | 11 |

220 | The number of ways in which one can post 5 letters in 6 post boxes ( mathbf{A} cdot 6^{5} ) B . 30 ( mathbf{c} cdot 5^{6} ) D. 11 | 11 |

221 | ( frac{C_{1}}{C_{0}}+2 cdot frac{C_{2}}{C_{1}}+3 cdot frac{C_{3}}{C_{2}}+dots+n cdot frac{C_{n}}{C_{n-1}}= ) A. ( frac{n(n+1)}{2} ) в. ( frac{n(n-1)}{2} ) c. ( frac{(n-1)(n+1)}{2} ) D. ( frac{n(n+2)}{2} ) | 11 |

222 | In how many ways we can select 3 letters of the word PROPORTION? | 11 |

223 | How many different words can be formed using all the letters of the word ‘ALLAHABAD’ when both ( L^{prime} s ) are not together A. 4200 B. 5812 ( c .6000 ) D. 5250 | 11 |

224 | How many 4 digit numbers are there which contain not more than 2 different digits? | 11 |

225 | The number of ways to rearrange the letters of the word CHEESE is A . 119 в. 240 ( c .720 ) D. 6 | 11 |

226 | How many numbers can be formed from the digits 1,3,5,9 if repetition of digits is not allowed? | 11 |

227 | 23. Aman X has 7 friends, 4 of an X has 7 friends, 4 of them are ladies and 3 are men. te Y also has 7 friends, 3 of them are ladies and 4 wah. Assume X and Y have no common friends. Then the total number of ways in which X and Y together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of X and Y are in this party, is: [JEEM 2017] [JEEM 20 (a) 484 (b) 485 (c) 468 (d) 469 | 11 |

228 | Each of 3 committees has 1 vacancy which is to be filled from a group of 6 people. Find the number of ways the 3 vacancies can be filled if Each person can serve on atmost 1 committee. | 11 |

229 | The value of ( x ) in the equation ( 3 x^{x+1} ) ( C_{2}=2 times^{x+2} C_{2}, x in N ) is ( mathbf{A} cdot x=4 ) B. ( x=5 ) c. ( x=6 ) D. ( x=7 ) | 11 |

230 | Find the number of words that can be formed by consider of thing all the possible permutations of the word FATHER. How many of these words begin with ( A ) and end with ( R ) | 11 |

231 | If ( boldsymbol{S}=^{404} boldsymbol{C}_{4}-^{4} boldsymbol{C}_{1} cdot^{303} boldsymbol{C}_{4}+^{4} ) ( C_{2} cdot^{202} C_{5}-^{4} C_{3} cdot^{101} C_{4}=(101)^{k} ) then ( k ) equal to A . 1 B . 2 ( c cdot 4 ) D. 6 | 11 |

232 | ( f^{n} C_{3}=^{n} C_{2}, ) then ( n ) is equal to ( A cdot 2 ) B. 3 ( c .5 ) D. None of these | 11 |

233 | How many different numbers of two digits can be formed with the digits 1,2,3,4,5,6 no digits being repeated? | 11 |

234 | A pod of 6 dolphins always swims single file, with 3 females at the front and 3 males in the rear. In how many different arrangements can the dolphins swim? A .24 B. 36 ( c .30 ) D. 18 | 11 |

235 | If different words are formed with all the letters of the word ‘AGAIN’ and are arranged alphabetically among themselves as in a dictionary, the word at the 50th place will be A . NAAGI c. IAAGN | 11 |

236 | ( sum_{0 leq i leq j leq 10}^{10} C_{j}^{j} C_{i} ) is equal to ( A cdot 3^{10} ) B. ( 3^{10}-1 ) ( c cdot 2^{10} ) D. ( 2^{10}-1 ) | 11 |

237 | A vehicle registration number consists of 2 letters of English alphabet followed by 4 digits, where the first digit is not zero. Then, the total number of vehicles with distinct registration numbers is A ( cdot 26^{2} times 10^{4} ) В. 26 ( P_{2} times ) 10 ( P_{4} ) c. ( 26 P_{2} times 9 times 10 P_{3} ) D . ( 26^{2} times 9 times 10^{3} ) | 11 |

238 | Find ( n, ) if ( ^{2 n} C_{3}:^{n} C_{2}=52: 3 ) | 11 |

239 | Which among the following is/are correct? This question has multiple correct options A. If an operations can be performed in ‘m’ different ways and a second operation can be performed in ‘n different ways, then both of these operations can be performed in ( ^{prime} m times n^{prime} ) ways together B. The number of arrangements of n different objects taken all at a times is n! C. The number of permutations of n different things taken ( r ) at a time, when each thing may be repeated any number of times is ( n ) D. The number of circular permutations of ‘n’ different things taken all at a time is ( frac{1}{2}(n-1) ! ). if clockwise and anticlockwise orders are taken as different. | 11 |

240 | 16. There are two urns. Urn A has 3 distinct red balls and urn B has 9 distinct blue balls. From each urn two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is [2010] (a) 36 (6) 66 (c) 108 (d) 3 | 11 |

241 | In how many ways three different rings can be worn in four fingers with at most one in each finger? | 11 |

242 | A password consists of two letters of alphabet followed by three digits chosen from 0 to ( 9 . ) Repeats are allowed. How many different possible passwords are allowed A .492804 B. 650000 c. 676000 D. 1757600 | 11 |

243 | Find the number of permutations that can be made with the letters of the word ( ‘ M O U S E^{prime} ) This question has multiple correct options A . ( 5 ! ) B. 5 c. 720 D. 120 | 11 |

244 | Ten different letters of an alphabet are given. Words with 5 letters are formed from these given letters. Then the number of words which have at least one letter repeated is A . 69760 B. 30240 c. 99748 D. None of these | 11 |

245 | A total of 28 handshakes was exchanged at the conclusion of a party. Assuming that each participant was equally polite towards all the others, the number of people present was: A . 14 B . 28 c. 56 D. 8 ( E .7 ) | 11 |

246 | Mario’s Pizza has 2 choices of crust: deep dish and thin-and-crispy. The restaurant also has a choice of 5 toppings: tomatoes, sausage, peppers, onions, and pepperoni. Finally, Mario’s offers every pizza in extra cheese as well as regular. If Linda’s volleyball team decides to order a pizza with 4 toppings. how many different choices do the teammates have at Mario’s Pizza? A ( cdot 24 ) в. 32 ( c cdot 28 ) D. 20 | 11 |

247 | Let ( a_{n} ) denote the number of all ( n ) -digit positive integers formed by the digits 0,1 or both such that no consecutive digits in them are ( 0 . ) Let ( b_{n}= ) the number of such ( n ) -digit integers ending with digit 1 and ( c_{n}= ) the number of such ( n ) -digit integers ending with digit 0. The value of ( b_{6} ) is ( A cdot 7 ) B. 8 ( c .9 ) D. 11 | 11 |

248 | Assertion If the postman delivers 1332 card to the students then number of students are 36. Reason If there are n students then total number of cards are ( n(n-1) ) A. Both (A) & (R) are individually true & (R) is correct explanation of (A), B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A) c. (A)is true but (R) is false D. (A)is false but (R) is true. | 11 |

249 | 9. A rectangle with sides of length (2m – 1) and (2n-1) units is divided into squares of unit length by drawing parallel lines as shown in the diagram, then the number of rectangles possible with odd side lengths is (2005) (a) (m+n-1)2 (c) m²n (b) 4m+n-1 (d) m(m+1)n(n+1) | 11 |

250 | Find the value of ( 4 ! ) | 11 |

251 | The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is A . 360 B. 192 ( c cdot 96 ) D. 48 | 11 |

252 | 7. 7. The sum C”%C2%). werel) – o if p<q is = 0 if p<q) is J, (where 10 (20) The sum im-i)' i=0 maximum when mis (a) 5 (b) 10 . (2002) 20 (d) | 11 |

253 | Find the sum of ( C_{0}+3 C_{1}+3^{2} C_{2}+ ) ( ldots+3^{n} C_{n} ) | 11 |

254 | 5 boys are to be arranged in a row. It two particular boys desire to sit in end places, the number of possible arrangements is A . 60 в. 120 c. 240 D. 12 | 11 |

255 | From 4 officers and 8 jawans, in how many ways can 6 be chosen such that to include exactly one officer. | 11 |

256 | The lock of a safe consists of five disce each of which features the digits. ( 0,1,2, ldots .9 . ) The safe can be opened by dialing a special combination og the digits. The number of days sufficient enough to open the safe, if the work day lasts ( 13 h ) and ( 5 s ) are needed to dial one combination of digits is: ( A cdot 9 ) B. 10 ( c cdot 11 ) D. 12 | 11 |

257 | Given s=1+q+q2 + …… +q”; 9+1) – 20 9 #1 Prove that Sy = 1+9 2 …..x/9+1n 1 2 7+1 24 2 n+1G + n+1C281 + n+1CzS2 + … + n+lCqSn = 2″ Sn (100 d e | 11 |

258 | There are two books of five volumes each and two books of two volumes each. In how many ways can these books be arranged in a shelf such that volumes of same books remain together? A . 38 B. 34 c. 56 D. 88 | 11 |

259 | If ( ^{n} C_{r-1}=left(k^{2}-8right)left(^{n+1} C_{r}right), ) then ( k ) belongs to This question has multiple correct options A ( cdot[-3,-2 sqrt{2}] ) (年 ( -[-3,-2 sqrt{2},-sqrt{2},-sqrt{2}] ) в. ( [-3,-2 sqrt{2}) ) c. ( 2 sqrt{2} .3 ) D. ( (2 sqrt{2}, 3 ) | 11 |

260 | Different words being formed by arranging the letters of the word “INTERMEDIATE”.All the words obtained are written in the form of a dictionary, If vowels ( & ) consonants occupy their original places, then the number of permutations is A ( cdot frac{6 !}{2 !} times frac{6 !}{2 !} ) в. ( frac{6 !}{3 !} times frac{6 !}{2 !} ) c. ( frac{6 ! times 6 !}{3 ! 2 ! 2 !} ) D. None of these | 11 |

261 | How many 6 -digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 and no digit appears more than once? | 11 |

262 | A five-digit numbers divisible by 3 is to be formed using the numerals 0, 1, 2, 3, 4 and 5, without repetition. The total number of ways this can be done is (1989-2 Marks) (a) 216 (b) 240 (c) 600 (d) 3125 | 11 |

263 | ( n P_{r} ) and ( n C_{r} ) are equal when: ( mathbf{A} cdot n=r ) B . ( n=r+1 ) ( mathbf{c} cdot r=1 ) D. ( n=r-1 ) | 11 |

264 | If ( ^{6} C_{n}+2^{6} C_{n+1}+^{6} C_{n+2}>^{8} C_{3} ),then the quadratic equations whose roots ( operatorname{are} alpha, beta ) and ( alpha^{n-1}, beta^{n-1} ) have A. 2 common root B. 1 common root c. no common roots D. imaginary roots | 11 |

265 | What is the sum of all 5 digit numbers which can be formed with digits 0,1,2 3, 4 without repetition A. 2599980 B. 2679980 c. 2544980 D. 2609980 | 11 |

266 | How many numbers consisting of 5 digits can be formed in which the digits 3,4 and 7 are used only once and the digit 5 is used twice A . 30 B. 60 c. 45 D. 90 | 11 |

267 | The number of positive integers which can be formed by using any number of digits from 0,1,2,3,4,5 without repetition A . 1200 в. 1500 c. 1600 D. 1630 | 11 |

268 | There are three stations ( A, B ) and ( C ) five routes for going from station ( boldsymbol{A} ) to station ( B ) and four routes for going from ( operatorname{station} B ) to station ( C ) Find the number of different ways through which a person can go from station ( A ) to ( C ) via ( B ) A . 10 B. 15 c. 20 D. 25 | 11 |

269 | If ( ^{n} boldsymbol{P}_{r}=mathbf{3 0 2 4 0} ) and ( ^{n} boldsymbol{C}_{r}=mathbf{2 5 2}, ) then the ordered pair ( (n, r) ) is equal to: A. (12,6) в. (10,5) c. (9,4) (年. ( 9,4,4) ) D. (16,7) | 11 |

270 | m men and n women are to be seated in a row so that no two women sit together. Ifm>n, then show that the number of m!(m +1)! ways in which they can be seated is (m-n+1)! | 11 |

271 | A road network as shown in the figure connect four cities. In how many ways can you start from any city (say A) and come back to it without travelling on the same road more than once ? ( A ) B. 12 ( c ) 0.16 | 11 |

272 | If PQRS is a convex quadrilateral with 3 4,5 and 6 points marked on side ( P Q, Q R ) RS and PS respectively. Then, the number of triangles with vertices on different sides is A .220 в. 270 ( c cdot 282 ) D. 342 | 11 |

273 | A school has 11 Maths teachers and 7 Science teachers. In how many ways can they be seated in a round table so that all the Science teachers do not sit together? ( mathbf{A} cdot 18 !-(12 ! times 7 !) ) B . ( 17 !-(12 ! times 7 !) ) c. ( 17 !-(11 ! times 7 !) ) D. ( 18 !-(11 ! times 7 !) ) | 11 |

274 | * cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and Tas are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover the card numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done is (JEE Adv. 2014) (a) 264 (6) 265 (6) 53 (d) 67 | 11 |

275 | We number both the rows and the columns of an ( 8 times 8 ) chess-board with the numbers 1 to ( 8 . A ) number of grains are placed onto each square, in such a way that the number of grains on a certain square equals the product of its row and column numbers. How many grains are there on the entire chessboard? A . 1296 B. 1096 ( c .2490 ) D. 1156 | 11 |

276 | The number of permutation of the letters of the word ( H I N D U S T A N ) such that neither the pattern ( ^{prime} boldsymbol{H} boldsymbol{I} boldsymbol{N}^{prime} ) nor ( ^{prime} D U S^{prime} ) nor ( ^{prime} T A N^{prime} ) appears, are : A . 166674 B. 169194 c. 166680 D. 181434 | 11 |

277 | 27. (0) Yually likely but not independent. If 12 identical balls are to be placed in 3 ident the probability that one of the boxes contains exac is : [JEE M 2015] te to be placed in 3 identical boxes, then boxes contains exactly 3 balls (2) 220 (7h (c) $ (3)” (6) 22 (1) ” (a) 55() | 11 |

278 | If the different permutations of all letters of the word ( B H A S K A R A ) are listed as in a dictionary, how many strings are there in this list before the first word starting with ( boldsymbol{B} ) ? | 11 |

279 | If ( ^{n} C_{r} ) denots the number of combinations of ( n ) things taken ( r ) at a time, then the expression ( ^{n} C_{r+1}+^{n} ) ( C_{r-1}+2 times^{n} C_{r} ) equals A. ( ^{n+2} C_{r} ) B. ( ^{n+2} C_{r+1} ) c. ( ^{n+1} C_{r} ) D. ( n+1 C_{r+1} ) | 11 |

280 | The number of even numbers with three digits such that if 3 is one of the digit then 5 is the next digit are A . 959 B. 285 c. 365 D. 512 | 11 |

281 | The number of ways in which 5 Mathematics, 4 Physics, 2 Chemistry books can be arranged in a shelf if the books on the same subject are kept together. A . 34,560 в. 34,500 c. 31,540 D. 32,850 | 11 |

282 | There are 2 gates to enter a school and 3 staircases from first floor to the second floor. How many possible ways are there for a student to go from outside the school to a classroom on the second floor and come back? | 11 |

283 | 10. If the letters of the word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number [2005] (a) 601 (b) 600 (c) 603 (d) 602 | 11 |

284 | Assertion The expression ( n !(10-n) ! ) is minimum for ( n=5 ) Reason The expression ( ^{2 m} C_{r} ) attains maximum value for ( boldsymbol{m}=boldsymbol{r} ) A. Both (A) & (R) are individually true & (R) is correct explanation of (A), B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A) c. (A)is true but (R) is false D. (A)is false but (R) is true | 11 |

285 | If ( ^{n} p_{4}=20 times^{n} p_{2} ) then find the value ( n ) ( A cdot 7 ) B. 5 ( c cdot 6 ) D. 4 | 11 |

286 | (n?)! is an integer 8. Prove by permutation or otherwise (n!” (nett). (2004 – 2 Marks) otobasic | 11 |

287 | Number of ways in which 3 boys and 3 girls can be seated on a line where two particular girls do not want to sit adjacent to a particular boy is equal to A . 36 B. 72 c. 144 D. 288 | 11 |

288 | How many words can be formed with the letters of the word ‘PARALLEL’ so that all L’s do not come together? | 11 |

289 | Number of permutation of 10 different objects taken all at a time in which particular 4 never comes together A. ( 10 ! times 4 ) B. ( 10 !-4 ) c. ( frac{7 ! 6 !}{4 !} ) D. ( 10 !-7 ! 4 ! ) | 11 |

290 | In how many ways 11 players can be selected from 15 players, if only 6 of these players can bowl and the 11 players must include atleast 4 bowlers? | 11 |

291 | In an examination there are three multiple choice questions and each question has 4 choices. Number of ways in which a student can fail to get all answer correct is? A . 11 B. 12 c. 27 D. 63 | 11 |

292 | 10. The value of COT® CO-2)*–20.3.) is where (2005) | 11 |

293 | In how many ways can the letter of the word SUCCESS be arranged so that (i) The two ( C^{prime} s ) are together bot no two ( S^{prime} s ) are together (ii) No two ( C^{prime} s ) and no two ( S^{prime} s ) are together. | 11 |

294 | In how many ways can 6 boys and 5 girls can be seated in a row such that no two girls are together? A . ( 6 ! 5 ) в. ( 2 times 5 ! 5 ) c. ( 6 ! times^{7} P_{5} ) D. ( 6 ! 6 ! ) | 11 |

295 | then equals (1998 – 2 Marks) n 1 te a.. = – r=on C (a) (n-1), b) nan 1) None of the above na | 11 |

296 | The product of five consecutive numbers is always divisible by? ( mathbf{A} cdot 60 ) B. 12 ( c cdot 120 ) D. 72 | 11 |

297 | If ( boldsymbol{x}, boldsymbol{y} in(mathbf{0}, boldsymbol{3} boldsymbol{0}) ) such that ( left[frac{boldsymbol{x}}{mathbf{3}}right]+left[frac{boldsymbol{3} boldsymbol{x}}{mathbf{2}}right]+ ) ( left[frac{boldsymbol{y}}{mathbf{2}}right]+left[frac{mathbf{3} boldsymbol{y}}{boldsymbol{4}}right]=frac{mathbf{1 1} boldsymbol{x}}{mathbf{6}}+frac{mathbf{5} boldsymbol{y}}{boldsymbol{4}}(text { where }[mathbf{x}] ) denote greatest integer ( leq x ) ) then the number of ordered pairs ( (x, y) ) is A . 10 B. 20 ( c cdot 24 ) D. 28 | 11 |

298 | In how many different ways can the letters of the word I KURUKSHETRA’ be arranged? | 11 |

299 | Let ( n ) be an odd integer greater than 1 and let ( c_{1}, c_{2}, cdots, c_{n} ) be integers. For each permutation ( boldsymbol{a}=left(boldsymbol{a}_{1}, boldsymbol{a}_{2}, cdots, boldsymbol{a}_{n}right) ) of ( (1,2, cdots, n) ) define ( S(a)=sum_{i=1}^{n} c_{i} a_{i} ) Prove that there exist permutations ( a neq b ) of ( {1,2, cdots, n} ) such that ( n ! ) is a divisor of ( mathrm{S}(mathrm{a})-mathrm{S}(mathrm{b}) ) A ( cdot frac{1}{2} n !(n !-1) equiv frac{1}{2}(n+1) ! sum_{i=1}^{n} c_{i}(bmod n !) . ) Because ( n>1 ) is odd, the right-hand side is congruent to 0 mod ( n ! ) while the lefthand side is not, a contradiction B ( cdot frac{1}{2} n !(n !+1) equiv frac{1}{2}(n+1) ! sum_{i=1}^{n} c_{0}(text { modn! }) . ) Because ( n>1 ) is odd, the right-hand side is congruent to 0 mod ( n ! ) while the lefthand side is not, a contradiction. c. ( frac{1}{2} n !(n !-1) equiv frac{1}{2}(n-1) ! sum_{i=1}^{n} c_{0}(text { modn! }) . ) Because ( n> ) is odd, the right-hand side is congruent to 0 mod ( n ! ) while the lefthand side is not, a contradiction. D. None of the above | 11 |

300 | If ( ^{12} C_{r+1}=^{12} C_{3 r-5}, ) find ( r ) | 11 |

301 | f ( (n+2) !=2550 times n ! ), find the value of ( n ) | 11 |

302 | There are 4 candidates for a Natural science scholarship, 2 for a Classical and 6 for a Mathematical scholarship,then find No. of ways these scholarships can be awarded is, A . 48 B. 12 c. 24 D. 8 | 11 |

303 | Let there are 4 red, 3 yellow and 2 green balls, then the total number of arrangements in a row A. 5 B. ( 4 ! ) ( c cdot 9 ) D. none of these | 11 |

304 | There are 5 letters and 5 addressed envelopes.The number of ways in which the letters can be placed in the envelopes so that none of them goes into the right envelope is A . 22 B. 44 ( c cdot 120 ) D. 119 | 11 |

305 | Assertion The expression ( ^{40} boldsymbol{C}_{r} .^{60} boldsymbol{C}_{0}+^{40} ) ( C_{r-1} .^{60} C_{1}+ldots ) attains maximum value when ( r=50 ) Reason ( ^{2 n} C_{r} ) is maximum when ( r=n ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Assertion is incorrect but Reason is correct | 11 |

306 | There are three coplanar parallel lines. If any p points are taken on each limes, the maximum number of triangles with vertices at these points is | 11 |

307 | Let ( X ) be a set containing n elements. Two subsets ( A ) and ( B ) of ( X ) are chosen at random, the probability that ( boldsymbol{A} cup ) ( B=X ) is A ( cdot frac{2 n}{2^{2 n}} ) в. ( frac{1}{2 n} C_{n} ) ( frac{c cdot frac{1 cdot 3 cdot 5 cdot ldots .(2 n-1)}{2^{n} n !}}{(2)} ) D. ( left(frac{3}{4}right)^{n} ) | 11 |

308 | In how many ways can the letters of the word ‘PERMUTATIONS’ be arranged if the (i) words start with ( P ) and ends with ( S ) and (ii) Vowels are all together. | 11 |

309 | Delegates from 9 countries include countries ( A, B, C, D ) are to be seated in a row. The number of possible seating arrangements, when the delegates of the countries ( A ) and ( B ) are to be seated next to each other and the delegates of the countries ( C ) and ( D ) are not to be seated next to each other is : A . 10080 B. 5040 c. 3360 D. 60480 | 11 |

310 | For ( 2 geq r geq n,left(begin{array}{c}r \ nend{array}right)+2left(begin{array}{c}r-1 \ nend{array}right)+left(begin{array}{c}r-2 \ nend{array}right) ) A ( cdotleft(begin{array}{c}r-1 \ n+1end{array}right) ) B ( cdot 2left(begin{array}{c}r+1 \ n+1end{array}right) ) ( mathbf{c} cdot 2left(begin{array}{c}r \ n+1end{array}right) ) D. ( left(begin{array}{c}r \ n+1end{array}right) ) | 11 |

311 | The number of different words that can be formed using all the letters of the letters of the word ( C )OCACOLA with start and ends with ( O ) is A. 840 в. 420 ( c cdot 120 ) D. 60 | 11 |

312 | The value of ( sum_{r=1}^{n} rleft(^{n} C_{r}+^{r} P_{r}right) ) is A ( cdot n cdot 2^{n-1}-1 ) B . ( n cdot 2^{n-1}+(n+1) ! ) C ( cdot n cdot 2^{n-1}+(n+1) !-1 ) D. ( n^{2}+n+5 ) | 11 |

313 | Assertion The value of ( left(begin{array}{c}51 \ 3end{array}right)+left(begin{array}{c}50 \ 3end{array}right)+left(begin{array}{c}49 \ 3end{array}right)+ ) ( left(begin{array}{c}48 \ 3end{array}right)+left(begin{array}{c}47 \ 3end{array}right)+left(begin{array}{c}47 \ 4end{array}right) ) is equal to ( left(begin{array}{c}52 \ 4end{array}right) ) Reason ( ^{n-1} C_{r}+^{n} C_{r+1}=^{n+1} C_{r+1} ) A. Both (A) & (R) are individually true & (R) is the correct explanation of (A), B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A). C. (A) is true but (R) is false, D. (A) is false but (R) is true. | 11 |

314 | 4 buses runs between Bhopal and Gwalior. If a man goes from Gwalior to Bhopal by a bus and comes back to Gwalior by another bus, then the total possible ways are A . 12 B. 16 ( c cdot 4 ) D. 8 | 11 |

315 | Find the number of ways in which the letters of the word ‘MUNMUN’ can be arranged so that no two alike letters are together? A . 30 B. 40 ( c cdot 60 ) D. 20 | 11 |

316 | 9. The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is [2004] (a) 8 C3 (6) 21 (c) 38 (d) 5 | 11 |

317 | A question paper consists of 11 question divided into two section 1 and 2. section 1 consits of 5 questions and section 2 consists of 6 questions. In how many ways can a student select 6 questions, taking at least 2 questions from each section? | 11 |

318 | A telegraph has 5 arms and each arm is capable of 4 distinct positions, including the position of first; what is the total number of signals that can be made? | 11 |

319 | From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. Then the number of such arrangement is: [2009] (a) at least 500 but less than 750 (b) at least 750 but less than 1000 (c) at least 1000 (d) less than 500 | 11 |

320 | A parallelogram is cut by two sets of ( boldsymbol{m} ) lines parallel to its sides. Find the number of parallelograms then formed. | 11 |

321 | The teacher distributes 6 pencils per student. Can you find how many pencils are needed for the given number of students(use ‘ ( z^{prime} ) for the number of students). | 11 |

322 | A has library has 6 copies of one book, 4 copies of each of three books and single copies of 8 books. Then number of arrangements of all the A ( cdot frac{(26) !}{(4 !)^{3} 6 !} ) В ( cdot frac{(26) !}{6 !(4 !)^{4}(6 !)^{3}} ) c. ( frac{(26) !}{6 !(4 !)^{2}(6 !)^{3}} ) D. ( frac{(26) !}{6 ! ! 4 ! 6 !} ) | 11 |

323 | The number of rational numbers ( frac{boldsymbol{p}}{boldsymbol{q}} ) where ( boldsymbol{p}, boldsymbol{q} in mathbf{1}, mathbf{2}, mathbf{3}, mathbf{4}, mathbf{5}, mathbf{6} ) is A . 23 B. 32 ( c cdot 36 ) D. 63 | 11 |

324 | ( (boldsymbol{k}-mathbf{1}) boldsymbol{C}_{(boldsymbol{k}-mathbf{1})}+^{boldsymbol{k}} boldsymbol{C}_{(boldsymbol{k}-mathbf{1})}+^{(boldsymbol{k}+mathbf{1})} boldsymbol{C}_{(boldsymbol{k}-mathbf{1})}+ ) ( +(n+k-2) C_{(k-1)}=? ) ( mathbf{A} cdot(n+k) C_{k} ) B. ( ^{(n+k+1)} C_{k} ) ( mathbf{C} cdot(n+k) C_{k-1} ) D. ( (n+k-1) C_{k} ) | 11 |

325 | 20. Let T, be the number of all possible triangles formed by joining vertices of an n-sided regular polygon. IfTm+1-T, = 10, then the value of n is: [JEE M 2013] . (a) 7 (b) 5 (c) 10 (d) 8 | 11 |

326 | There are ‘mn’ letters and n post boxes. The number of ways in which these letters can be posted is: A ( cdot(m n)^{n} ) B . ( (m n)^{m} ) ( mathrm{c} cdot m^{m} ) D. ( n^{m} ) | 11 |

327 | If ( frac{n}{^{n} C_{r}+4^{n} C_{r+1}+6^{n} C_{r+2}+4^{n} C_{r+3}+^{n} C} ) ( frac{boldsymbol{r}+boldsymbol{k}}{boldsymbol{n}+boldsymbol{k}}, ) then the value of ( boldsymbol{k} ) equals A . 1 B. 2 ( c cdot 4 ) D. None of these | 11 |

328 | The number of different signals that can be formed by using any number of flags from 4 flags of different colours is A .24 B. 256 ( c cdot 64 ) D. 60 | 11 |

329 | The total number of integers n such that ( 2 leq n leq 2000 ) and the H.C.F of ( n ) and 36 is equal to 1 is A . 665 в. 666 ( c cdot 667 ) D. None of these | 11 |

330 | Consider the letters of the word INTERMEDIATE. Number of words formed by using all the letters of the given words are | 11 |

331 | If the four letter words(need not be meaningful) are to be formed using the letters from the word “MEDI TERRANEAN” such that the first letter is | 11 |

332 | If ( ^{n} C_{8}=^{n} C_{27}, ) then what is the value of ( boldsymbol{n} ? ) A . 35 B. 22 c. 28 D. 41 | 11 |

333 | Determine ( n ) if ( ^{2 n} C_{3}:^{n} C_{3}=12: 1 ) | 11 |

334 | ( mathrm{IF}^{n} boldsymbol{C}_{r}=^{n} boldsymbol{P}_{r} ) then ( r ) can be A . B. 1 ( c cdot 3 ) D. Either(1) or (2) | 11 |

335 | There are 20 points on a plane, with no 3 being collinear. The number of triangles that can be formed by connecting the points is A . 1140 в. 940 c. 380 D. 220 | 11 |

336 | In how many ways can 6 women draw water from 6 wells, if no well remains used? | 11 |

337 | The rank of the word ( N U M B E R ) obtained, if the letters of the word ( N U M B E R ) are written in all possible orders and these words are written out as in a dictionary is A .468 в. 469 c. 470 D. 471 | 11 |

338 | 5. For 2 <rsn, + 1 (2000) For 2575 n. ()+2(„”.) +(",)- zobas, @ (m) o (:) *** (*) | 11 |

339 | If ( r, s ) and ( t ) are prime numbers and ( p, q ) are positive integers such that the LCM of ( p, q ) is ( r^{2} t^{4} s^{2} ) then the number of ordered pair ( (p, q) ) is A . 254 B. 252 c. 225 D. 224 | 11 |

340 | If ( m=^{n} mathrm{C}_{2}, ) then ( ^{m} mathrm{C}_{2} ) equals ( mathbf{A} cdot^{n+1} mathbf{C}_{4} ) B. ( 3 x^{n+1} mathrm{C}_{4} ) ( mathrm{c} cdot^{n} mathrm{C}_{4} ) D. ( n+1 mathrm{C}_{3} ) | 11 |

341 | A batsman can score 0,1,2,3,4 or 6 runs from a ball. The number of different sequences in which he can score exactly 30 runs in an over of six balls is: ( A cdot 4 ) B. 72 ( c .56 ) D. 7 | 11 |

342 | The greatest possible number of points of intersection of 9 different straight lines and 9 different circles in a plane is: A . 117 B. 153 c. 270 D. none of these | 11 |

343 | The number of 5 digit telephone numbers having least one of their digits repeated is A .90,000 в. 100,000 c. 30,240 D. 69,760 | 11 |

344 | The number of all the possible selection which a student can make for answering one or more questions out of eight given question in a paper, which each question has an alternative is A . 255 B. 6560 ( c .6561 ) D. none of these | 11 |

345 | Find the number of arrangements of the letters of the word INDEPENDENCE. In how many of these arrangements do all vowels occur together. | 11 |

346 | If some or all of ( n ) things be taken at a time, prove that the number of combinations is ( 2^{n}-1 ) | 11 |

347 | In how many ways can 13 question papers be arranged so that the best and the worst are never come together? A . ( 12 ! times 11 ) В. ( 11 times 12 ! ) c. ( 12 !^{2} ) D. none of these | 11 |

348 | The numbers of ways in whi | 11 |

349 | The number of positive integral solutions of the equation ( boldsymbol{x}_{1} boldsymbol{x}_{2} boldsymbol{x}_{3} boldsymbol{x}_{4} boldsymbol{x}_{5}=mathbf{1 0 5 0} ) is A. 1800 B. 1600 ( c cdot 1400 ) D. 1875 | 11 |

350 | The total number of different combinations of one or more letters which can be made from the letter of the word MISSISSIPPI is, A. 150 в. 148 ( c cdot 149 ) D. 146 | 11 |

351 | The number of ways of distributing 12 identical oranges among 4 children so that every child gets at least one orange and none of the child gets more than 4 is A. 3 B. 52 ( c .35 ) D. 42 | 11 |

352 | ( ^{10} C_{3}=^{10} C_{2 r} ) then ( r=? ) | 11 |

353 | A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him [2003] (a) 346 (b) 140 (c) 196 (d) 280 is | 11 |

354 | There are n points on a circle. The number of straight lines formed by joining them is equal to ( A cdot^{n} C_{2} ) в. ( ^{n} P_{2} ) ( mathrm{c} cdot^{n} C_{2}-1 ) D. None of these | 11 |

355 | How many different words can be fanned by jumbling the letters of the word MISSISSIPPI in which no two ‘s’ are adjacent A ( cdot 7 cdot^{6} C_{2} cdot^{8} C_{4} ) B . ( 6 cdot 7 cdot^{8} C_{4} ) ( mathbf{c} cdot 6 cdot 8 cdot^{7} C_{4} ) D. ( 8 cdot^{6} C_{4} cdot^{7} C_{4} ) | 11 |

356 | The number of distinct terms in ( (a+ ) ( boldsymbol{b}+boldsymbol{c}+boldsymbol{d}+boldsymbol{e})^{3} ) is A . 35 B . 38 ( c cdot 42 ) D. 45 | 11 |

357 | The number of six-digit numbers that can be formed from the digits 1,2,3,4,5,6 and 7 so that digits do not repeat and the terminal digits are even is A .144 B. 72 c. 288 D. 720 | 11 |

358 | The total number of ( 9- ) digit numbers of different digits is A. ( 10(9 !) ) в. ( 8(9 !) ) c. ( 9(9 !) ) D. none of these | 11 |

359 | 11. The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is (2007-3 marks) (a) 360 (b) 192 (0) 46 (d) 48 | 11 |

360 | The least positive integral value of ( x ) which satisfies the in equality ( ^{10} C_{x-1}>2 .^{10} C_{x} ) is A. 7 B. 8 c. 9 D. 10 | 11 |

361 | A letter lock consists of three rings each are marked with 10 different letters. Find the number of ways in which it is possible to make an unsuccessful attempts to open the lock. | 11 |

362 | In how many different ways can the letters of the word ( ^{prime} L E A D I N G^{prime} ) be arranged such that all the consonants are together? ( mathbf{A} cdot 576 ) в. 625 ( c cdot 676 ) D. 720 | 11 |

363 | A group consists of 4 couples in which each of the 4 persons have one wife each. In how many ways could they be arranged in a straight line such that the men and women occupy alternate positions? A . 1152 в. 1278 ( c cdot ) 1296 D. 1176 | 11 |

364 | In how many ways can 3 prizes be distributed among 4 boys, when (i) no boy gets more than one prize? (ii) a boy may get member of prizes? (iii) no boy gets all the prizes? | 11 |

365 | The maximum number of persons in a country in which no two persons have an identical set of teeth assuming that there is no person without a tooth is ( mathbf{A} cdot 2^{3} ) B. ( 2^{32}-1 ) ( c .32 ! ) D. 32! – 1 | 11 |

366 | A man has 9 friends, 4 boys and 5 girls. In how many ways can be invite them, if there have to be exactly three girls in the invites? A . 320 в. 160 c. 80 D. 200 | 11 |

367 | There are 5 doors to a lecture hall. The number of ways that a student can enter the hall and leave it by a different door is A . 20 B. 16 c. 19 D. 2 | 11 |

368 | If ( ^{n} C_{3}=^{n} C_{9}, ) then ( ^{n} C_{2}= ) A . 66 B. 132 c. 72 D. 98 | 11 |

369 | Three players play a total of 9 games. In each game, one person wins and the other two lose; the winner gets 2 points and the losers lose 1 each. The number of ways in which they can play all the 9 games and finish each with a zero score is ( mathbf{A} cdot 84 ) B. 1680 c. 7056 D. | 11 |

370 | How many 6-letter words with distinct letters in each can be formed using the letters of the word EDUCATION? How many of these begin with I A. ( ^{9} P_{6^{prime}}^{8} P_{5} ) в. ( ^{9} P_{6^{prime}}^{9} P_{5} ) ( mathrm{c} cdot^{8} P_{6^{prime}}^{8} P_{5} ) D. ( ^{8} P_{6^{prime}}^{8} P_{4} ) | 11 |

371 | Number of increasing permutations of ( m ) symbols are there from the ( n ) set numbers ( left{a_{1}, a_{2}, ldots, a_{n}right} ) where the order among the numbers is given by ( boldsymbol{a}_{1}<boldsymbol{a}_{2}<boldsymbol{a}_{3}<ldots boldsymbol{a}_{n-1}<boldsymbol{a}_{n} ) is : | 11 |

372 | The digits 4,5,6,7 and 8 written in every possible order. The number of numbers greater than 56,000 is ( mathbf{A} cdot 78 ) B. 72 c. 90 D. 88 | 11 |

373 | The total number of flags with three horizontal strips in order, which can be formed using 2 identical red, 2 identical green and 2 identical white strips is equal to A. ( 4 ! ) в. ( 3 times(4 !) ) c. ( 2 times(4 !) ) D. None of these | 11 |

374 | The difference between the different permutation of the word BANANA and rank of the word BANANA is A. 60 B. 35 ( c cdot 24 ) D. None of these | 11 |

375 | 1. Total number of four digit odd numbers that can be formed using 0, 1, 2, 3, 5, 7 (using repetition allowed) are [2002] (a) 216 (b) 375 (c) 400 (d) 720 | 11 |

376 | If ( r>1, ) then ( frac{n P_{r}}{n C_{r}} ) is This question has multiple correct options A. is an integer B. may be fraction c. is an odd number D. an even number | 11 |

377 | Two persons entered a Railway compartment in which 7 seats were vacant.The number of ways in which they can be seated is A . 30 B. 42 ( c cdot 720 ) D. 360 | 11 |

378 | Solve the following inequalities. ( C_{m-2}^{13}>C_{m}^{18}, m epsilon N ) | 11 |

379 | How many 6 digits odd numbers greater than 60,0000 can be formed from the digits 5,6,7,8,9,0 if Repetitions are not allowed: A . 60 B. 120 c. 240 D. 480 | 11 |

380 | The product of ( r ) consecutive integers is divisible by ( r ! ) A. True B. False | 11 |

381 | Suppose a lot of ( n ) objects having ( n_{1} ) objects of one kind, ( n_{2} ) objects are of second kind, ( n_{3} ) objects of third kind, ( dots ) ( n_{k} ) objects of ( k^{t h} ) kind satisfying the condition ( n_{1}+n_{2} dots+n_{k}=n, ) then the number of possible arrangements/permutation of ( mathrm{m} ) objects out of this lot is the coefficient of ( x^{m} ) in the expansion of ( m ! prodleft{sum_{lambda=0}^{a_{1}} frac{x^{lambda}}{lambda !}right} ) The number of permutations of the letters of the word SURITI taken 4 at a time is A . 360 B. 240 c. 216 D. 192 | 11 |

382 | Two classrooms A and B having capacity of 25 and ( (n-25) ) seats respectively. ( A_{n} ) denotes the number of possible seating arrangements of room ( A^{prime}, ) when ‘n’ students are to be seated in these rooms, starting from room ( ^{prime} boldsymbol{A}^{prime} ) which is to be filled up to its capacity. If ( boldsymbol{A}_{n}-boldsymbol{A}_{n-1}=mathbf{2 5 !}left(^{49} boldsymbol{C}_{25}right) ) then ‘n’ equals: A . 50 B . 48 c. 49 D. 51 | 11 |

383 | Assertion ( sum_{r=0}^{100}left(^{500-r} C_{3}right)+^{400} C_{4}=^{501} C_{4} ) Reason ( boldsymbol{n} boldsymbol{C}_{boldsymbol{r}}+^{boldsymbol{n}} boldsymbol{C}_{boldsymbol{r}-boldsymbol{1}}=boldsymbol{n}+boldsymbol{1} boldsymbol{C}_{boldsymbol{r}} ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion C. Assertion is correct but Reason is incorrect D. Assertion is incorrect but Reason is correct | 11 |

384 | The coefficients of ( x^{p} ) and ( x^{q} ) (p and q are positive integers) in the expansion of ( (1+x)^{p+q} ) are A. Equal B. Equal with opposite signs c. Reciprocals to each other D. None of these | 11 |

385 | State following are True or False If ( m=n=p ) and the groups have identical qualitative characterstic then the number of groups ( =frac{(3 n) !}{n ! n ! n ! 3 !} ) Note : If 3n different things are to be distributed equally three people then the number of ways ( =frac{(3 n) !}{(n !)^{3}} ) A . True B. False | 11 |

386 | ( operatorname{Let}left(a_{1}, a_{2}, a_{3}, dots, a_{2011}right) ) be a permutation (that is a rearrangement) of the numbers ( 1,2,3, ldots, 2011 . ) Show that there exist two numbers ( j, k ) such that ( 1 leq j<k leq 2011 ) and ( left|a_{j}-jright|= ) ( left|boldsymbol{a}_{boldsymbol{k}}-boldsymbol{k}right| ) A. 20101005 B. 2011 1006 c. 20111005 D. 2011 1010 | 11 |

387 | a) How many different words can be formed using the letters of the word HARYANA? b) How many of these begin with H and end with N? c) In how many of these ( mathrm{H} ) and ( mathrm{N} ) are together? | 11 |

388 | The number of integer solutions ( (x, y, z) ) of ( x y z=18 ) is A . 48 B. 64 c. 72 D. 81 | 11 |

389 | A committee of 12 is to be formed from nine women and eight men. In how many ways this be done if at least five women have to be included in a committee? In how many of these committees a. the women hold majority? b. the men hold majority? | 11 |

390 | The value of ( sum_{r=0}^{n-1}^{n} C_{r} /left(^{n} C_{r}+^{n} C_{r+1}right) ) equals ( mathbf{A} cdot n+1 ) в. ( n / 2 ) ( mathbf{c} cdot n+2 ) D. none of these | 11 |

391 | 3. The value of the expression 47 Cz is equal to (1982 – 2 Marks) (b) 52C, (d) none of these (c) 526 | 11 |

392 | There are 12 intermediate stations between two places ( A ) and B. Find the number of ways in which a train can be made to stop at 4 of these intermediate stations so that no two stopping stations are consecutive | 11 |

393 | The value of ( 3^{7} C_{0}+4^{7} C_{1}+5^{7} C_{2}+ ) ( ———–10^{7} C_{7} ) is A ( cdot 10(2)^{6} ) B . ( 13(2)^{7} ) c. ( 14(2)^{6} ) D. ( 13(2)^{6} ) | 11 |

394 | A three-digit code for certain locks uses the digits 0,1,2,3,4,5,6,7,8,9 according to the following constraints. The first digit cannot be 0 or 1 the second digit must be 0 or ( 1, ) and the second and third digits cannot both be 0 in the same code. How many different codes are possible? A. 144 в. 152 c. 160 D. 168 E. 176 | 11 |

395 | Let ( A=(x mid x ) is a prime number and ( x ) the number of different rational numbers, whose numerator and denominator belong to ( boldsymbol{A} ) is: ( A cdot 91 ) B. 84 ( c cdot 106 ) D. None of these | 11 |

396 | A five-digit number divisible by 3 is to be formed using the digits 0,1,2,3 and 4 without repetition of digits. What is the number of ways this can be done? ( mathbf{A} cdot 96 ) B. 48 ( c cdot 32 ) D. No number can be formed | 11 |

397 | How many numbers of three digits can be formed using the digits 1,2,3,4,5 without repetition of digits is x. How many of these are even is y.Find ( mathbf{x}+mathbf{y} ) | 11 |

398 | A total of 324 coins of 20 paise and 25 paise make a sum of ( R s .71 ), the number of 25 paise coins is A . 124 в. 140 c. 200 D. 210 | 11 |

399 | If the number of selections of 3 difference letters from the word ( boldsymbol{S U M} boldsymbol{A} boldsymbol{N} ) | 11 |

400 | Given: ( frac{20 !}{18 !}=380 ) A. True B. False c. Either D. Neither | 11 |

401 | The total number of all the numbers divisible by ( 5, ) lying between 4000 and 5000 and can be formed using the digits 4,5,6,7 and 8 is : A . 125 B. 9 c. 25 D. 625 | 11 |

402 | If all the permutations of the letters in the word ‘OBJECT’ are arranged (and numbered serially) in alphabetical order as in a dictionary, then the ( 717^{t h} ) word is A. толесв в. тоЕлвс c. тосле D. толсве | 11 |

403 | If ( 4 .^{n} C_{6}=33 .^{n-3} C_{3} ) then ( n ) is equal to ( mathbf{A} cdot mathbf{9} ) B. 10 c. 11 D. none of these | 11 |

404 | How many total distinct terms are there in the expansion of ( (x+y+z+t)^{10} ) | 11 |

405 | Six X’s have to be placed in the square of given figure such that each row contains at least one X. Then number of ways doing so are A . 28 B. 26 ( c .6 ) D. ( 8 ! .6 ) | 11 |

406 | State whether the statement is true/false. An die is tossed twice. Find the | 11 |

407 | The number of different 7 -digit telephone numbers that can be formed by using 0,1,2,3,4,5,6,7,8,9 is : A ( cdot 10^{7} ) B. ( 9 times 10^{6} ) ( c .9 ! ) D. None of these | 11 |

408 | In a packet there are m different books, n different pens and p different pencils. The number of selections of at least one article of each type from the packet is A ( cdot 2^{m+n+p}-1 ) B . ( (m+1)(n+1)(p+1)-1 ) ( mathbf{c} cdot 2^{m+n+p} ) D – ( left(2^{m}-1right)left(2^{n}-1right)left(2^{p}-1right) ) | 11 |

409 | Three persons entered a railway compartment in which 5 seats were vacant. Find the number of ways in which they can be seated A . 30 B. 45 ( c cdot 120 ) D. 60 | 11 |

410 | Given that (1979) 9 +2C2x+3C x2 + ………….. + 2n C x2n-1 = 2n (1 + x)2n- (2n)! r=0,1,2,…………, 2n where C,= Pr!(2n-r)! Prove that C,2-20,2+3C,2…. 2nC, 2=(-1)”n C – 2N2n | 11 |

411 | A letter lock consists of three rings each marked with fifteen different letters; find in how many ways it is possible to make an unsuccessful attempt to open the lock. | 11 |

412 | Find the number of 4 letter words that can be formed using the letters of the word PISTON, in which at least one letter is repeated. | 11 |

413 | For a chess tournament 13 people were selected for quarter finals. Each person plays two matches with the other. How many matches have been held in the whole tournament? A .144 в. 156 ( c .185 ) D. 116 | 11 |

414 | How many arrangements of four ( 0^{prime} s ) (zeroes), two ( 1^{prime} s ) and two ( 2^{prime} s ) are there in which the first 1 occur before the first 2? A . 420 в. 360 ( c cdot 310 ) D. 210 | 11 |

415 | Find the value of ( frac{mathbf{6} !}{mathbf{3} !} ) | 11 |

416 | Number of words which begins with a vowel and ends with a consonant by permuting the letters of the word HARSHITA” A . 2340 в. 2700 ( c .1800 ) D. 1980 | 11 |

417 | There are 8 men and 10 women and you need to form a committee of 5 men and 6 women. In how many ways can the committee be formed? A . 10420 B. 11420 c. 11760 D. None of these | 11 |

418 | Arrange the following values of ( n ) in ascending order. ( boldsymbol{A}:^{n} boldsymbol{P}_{5}=^{n} boldsymbol{P}_{6} Rightarrow boldsymbol{n}= ) ( boldsymbol{B}:^{boldsymbol{n}} boldsymbol{P}_{12}=^{boldsymbol{n}} boldsymbol{P}_{boldsymbol{8}} Rightarrow boldsymbol{n}= ) ( boldsymbol{C}:^{n} boldsymbol{C}_{(boldsymbol{n}-mathbf{3})}=mathbf{1 0} Rightarrow boldsymbol{n}= ) ( boldsymbol{D}:^{(boldsymbol{n}+1)} boldsymbol{P}_{5}:^{boldsymbol{n}} boldsymbol{P}_{6}=mathbf{1}: boldsymbol{2} Rightarrow boldsymbol{n}= ) ( A ). САВ D в. САDВ c. АСВD D. DBAC | 11 |

419 | Three horses ( boldsymbol{H}_{1}, boldsymbol{H}_{2}, boldsymbol{H}_{3} ) entered a field which has seven portions marked ( P_{1}, P_{2}, P_{3}, P_{4}, P_{5}, P_{6} ) and ( P_{7} ). If no two horses are allowed to enter the same portion of the field, in how many ways can the horses graze the grass of the field? | 11 |

420 | A letter Lock consists of three rings each marked with 5 different letters Number of maximum attempts to open the lock is: A . 124 в. 125 c. 120 D. 75 | 11 |

421 | A letter lock consists of 4 rings, each ring contains 9 non-zero digits. This lock can be opened by setting a 4 digit code with the proper combination of each of the 4 rings.Maximum how many codes can be formed to open the lock? | 11 |

422 | A seven-digit number without repetition and divisible by 9 is to be formed by using seven digits out of ( mathbf{1}, mathbf{2}, mathbf{3}, mathbf{4}, mathbf{5}, mathbf{6}, mathbf{7}, mathbf{8}, mathbf{9} . ) The number of ways in which this can be done is A . 9 в. ( 2(7 !) ) c. ( 4(7 !) ) D. None of these | 11 |

423 | If ( ^{18} boldsymbol{C}_{boldsymbol{r}}=^{18} boldsymbol{C}_{boldsymbol{r}+mathbf{2}}, ) find ( ^{r} boldsymbol{C}_{boldsymbol{5}} ) | 11 |

424 | In a polygon, no three diagonals are concurrent. If the total number of points of intersection of diagonals interior to the polygon is 70 then the number of diagonals of the polygon is | 11 |

425 | 3. Five digit number divisible by 3 is formed using 0, 1, 2, 3, 4 and 5 without repetition. Total number of such numbers are [2002] (a) 312 (6) 3125 (c). 120 (d) 216 | 11 |

426 | In how many ways can the letters of the word COMBINE be arranged so as to begin and end with a vowel? Also find the number of words that can be formed without changing the relative order of the vowels and consonants. | 11 |

427 | The number of unsuccessful attempts that can be made by a thief to open a number lock having 3 rings in which each rings contains 6 numbers is A . 205 B. 200 c. 210 D. 215 | 11 |

428 | A new flag is to be designed with six vertical strips using some or all of the colour yellow, green, blue and red. Then, the number of ways this can be done such that no two adjacent strips have the same colour is? A. ( 12 times 81 ) B . ( 16 times 192 ) c. ( 20 times 125 ) D. ( 24 times 216 ) | 11 |

429 | Find ( r ) if ( ^{15} C_{3 r}=^{15} C_{r+3} ) | 11 |

430 | The value of ( ^{10} boldsymbol{C}_{4}+^{9} boldsymbol{C}_{4}+^{8} boldsymbol{C}_{4}+ldots+^{5} ) ( C_{4} ) is ( -dots ) A. ( 11 C_{5} ) B. ( ^{11} C_{4} ) c. ( ^{11} C_{7} ) D. ( ^{11} C_{5}-1 ) | 11 |

431 | If the four letter words (need not to meaningful) are to be formed using the letter from the word “MEDITERRANEAN” such that the first letter is ( R ) and the fourth letter is ( mathrm{E} ), then the total number of all such words is: A . 110 B . 59 c. ( frac{11 !}{(2 !)^{3}} ) D. 56 | 11 |

432 | The number of words that can be formed using any number of letters of the word “KANPUR” without repeating any letter is A . 720 в. 1956 ( c .360 ) D. 370 | 11 |

433 | Number of odd numbers of five distinct digits can be formed by the digits ( mathbf{0}, mathbf{1}, mathbf{2}, mathbf{3}, mathbf{4}, ) is A .24 в. 120 c. 48 D. 36 6 | 11 |

434 | Find the number of four digit numbers that are divisible by 15 and formed with the digits 0,1,2,3,4,5 when repetition is not allowed. A . 25 B. 34 c. 40 D. 55 | 11 |

435 | 8064 is resolved into all possible product of two factors. Find the number of ways in which this can be done? A .24 B . 21 c. 20 D. None of these | 11 |

436 | The exponent of 5 in ( ^{120} C_{60}, ) is ( mathbf{A} cdot mathbf{1} ) B. ( c cdot 2 ) D. 3 | 11 |

437 | The number of arrangements in which the letters of the word DEEDS be arranged such that neither the two D’s nor the two E’s are together is A . 10 B. 12 c. 18 ( D ) | 11 |

438 | 11. At an election, a voter may vote for any number of candidates, not greater than the number to be elected. There are 10 candidates and 4 are of be selected, if a votér votes for at least one candidate, then the number of ways in which he can vote is [2006] (a) 5040 (6) 6210 (C) 385 (d) 1110 | 11 |

439 | The number of four digit numbers that can be formed from the digit 0,1,2,3,4,5 with at least one digit repeated is ? A . 420 в. 560 ( c .750 ) D. None | 11 |

440 | Prove that ( n_{r+1}+2^{n} C_{r}+^{n} C_{r-1}= ) | 11 |

441 | 21. The number of integers greater than o, Negers greater than 6,000 that can be formed, using the digits 3,5, 6, 7 and 8, without repetition, is : [JEE M 2015] (a) 120 (6) 72 (c) 216 . (d) 192 | 11 |

442 | A man has 7 relatives, 4 of them are ladies and 3 gentlemen; his wife has 7 relatives and 3 of them are ladies and 4 gentlemen. The number of ways they can invite a dinner party of 3 ladies and 3 gentlemen so that there are 3 of men’s relatives and 3 of wife’s relative is A . 455 B. 565 c. 485 D. None of these | 11 |

443 | The number of ways in which 6 rings can be worn on the four fingers of one hand is ( mathbf{A} cdot 4^{6} ) в. ( ^{6} C_{4} ) ( c cdot 6^{4} ) D. None of these | 11 |

444 | For ( mathbf{0} leq boldsymbol{r} leq boldsymbol{n},^{boldsymbol{n}} boldsymbol{C}_{boldsymbol{r}}= ) | 11 |

445 | Sabnam has 2 school bags, 3 tiffin boxes and 2 water bottles.show in how many ways can she carry these item | 11 |

446 | How many words can be formed from the word “BHARAT” A . 360 B. 180 ( c cdot 90 ) D. 45 | 11 |

447 | On the occasion of Dipawali festival each student of a class sends greeting cards to one another. If the postmen deliver 1640 greeting cards to the students of this class, then the number of students in the class is A . 39 B. 41 ( c cdot 5 ) D. 53 | 11 |

448 | Number of 4 digit numbers of the form ( N=a b c d ) which satisfy following three conditions (i) ( 4000 leq N<6000 ) (ii) ( N ) is multiple of 5 (iii) ( 3 leq b<c leq 6 ) is equal to A . 12 B . 18 ( c cdot 24 ) D. 48 | 11 |

449 | How many arrangements of the word ( A R R A N G E ) can be made ( (i) ) If the two ( R^{prime} s ) are not allowed to come together, let it be ( k ? ) ( (i i) ) If neither the two ( R^{prime} s ) nor the two ( A^{prime} s ) are allowed to come together, let it be ( mathrm{m} ) Find the sum of digits of ( mathrm{m}+mathrm{k} ) ? | 11 |

450 | The number of numbers greater than ( 10^{6} ) that can be formed using the digits of the number ( 2334203, ) if all the digits of the given number must be used, is | 11 |

451 | 1. If C, stands for “C,, then the sum of the series –[C3 – 2C + 3C – + (-1)” (n + 1)C71, n! where n is an even positive integer, is equal to (1986 – 2 Marks) (a) 0 (b) (-1)1/2 (n+1) (c) (–1)n/2(n+2) (d) (-1)”n (e) none of these. | 11 |

452 | Evaluate: ( mathbf{1}^{2} boldsymbol{c}_{1}+mathbf{2}^{2} boldsymbol{c}_{2}+mathbf{3}^{2} boldsymbol{c}_{3}+boldsymbol{4}^{2} boldsymbol{c}_{4}+ldots ldots+boldsymbol{n}^{2} boldsymbol{c}_{n} ) | 11 |

453 | The exponent of 18 in ( 200 !, ) is ( mathbf{A} cdot 24 ) B. 46 c. 47 D. 48 | 11 |

454 | Given 5 different green dyes, four different blue dyes and three different red dyes, the number of combinations of dyes which can be chosen taking at least one green and one blue dye is?? A . 3600 B. 3720 ( c .3800 ) D. 3500 | 11 |

455 | Number of 9 digits numbers divisible by nine using the digits from 0 to 9 if each digit is used almost once is ( boldsymbol{K}=mathbf{8 !} ) then ( K ) has the value equal to | 11 |

456 | The number of nine digit numbers that can be formed with different digits is A. 9.8 B. ( 8.9 ! ) c. ( 9.9 ! ) D. 10 | 11 |

457 | Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First the women choose the chairs from amongst the chairs marked 1 to 4 ; and then the men select the chairs from amongst the remaining. The number of possible arrangements is (1982 – 2 Marks) (b) 4P2x4P3 (a) C3 x 4C (e) 4c2+Pg (d) none of these | 11 |

458 | A teacher of a class wants to set one question from each of two exercises in a book. If there are 15 and 12 questions in the two exercises respectively, then in how many ways can the two questions be selected? | 11 |

459 | The number of ways in which one or more letters be selected from the letters ( A A A A B B C C C D E F ) is A . 476 в. 487 c. 435 D. 47 | 11 |

460 | ( boldsymbol{C}_{3} / boldsymbol{4}+boldsymbol{C}_{5} / boldsymbol{6}+boldsymbol{C}_{7} / boldsymbol{8}+ldots ldots= ) ( ^{mathbf{A} cdot frac{2^{n+1}-n^{2}-n-2}{2(n+1)}} ) ( ^{text {В } cdot frac{left.2^{n+2}+n^{2}+n-2right)}{n+1}} ) c. ( frac{left(3^{n+2}-n^{2}-n-2right)}{n+1} ) D. None of these | 11 |

461 | How many ( 3- ) digit numbers can be formed by using the digits 0,1,3,5,7 while each digit may be repeated any number of times? | 11 |

462 | The sum of integers from 1 to 100 that are divisible by 2 or is [2002] (a) 3000 (b). 3050 (c) 3600 (d) 3250 | 11 |

463 | Four alphabets ( mathrm{E}, mathrm{K}, mathrm{S} ) and ( mathrm{V}, ) one in each, were purchased from a plastic warehouse. How many ordered pairs of alphabets, to be used as initials, can be formed from them? | 11 |

464 | How may words can be formed using the letter ( A ) thrice, the letter ( B ) twice and the letter C once? ( mathbf{A} cdot 60 ) в. 120 c. 90 D. 59 | 11 |

465 | In a crossword puzzle, 20 words are to be guessed of which 8 words have each an alternative solution also. The number of possible solutions will be A. ( 20 P_{8} ) в. ( 20 mathrm{C}_{8} ) c. 512 D. 256 | 11 |

466 | The number of ways in which a mixed doubles tennis game can be arranged between 10 players consisting of 6 men and 4 women is A . 180 B. 90 ( c cdot 48 ) ( D cdot 12 ) | 11 |

467 | How many different nine-digit numbers can be formed from the digits of the number 223355888 by rearrangement of the digits so that the odd digits occupy even places? This question has multiple correct options | 11 |

468 | In how many ways we can select 5 cards from a deck of 52 cards, if each selection must include atleast one king | 11 |

469 | Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have at least one letter repeated are (1982 – 2 Marks) (a) 69760 (b) 30240 (c) 99748 (d) none of these | 11 |

470 | A candidate is required to answer 6 out of 10 questions which are divided into two groups each containing 5 questions and he is not permitted to attempt more than 4 from each group. In how many ways can he make up his choice? | 11 |

471 | The value of the expression ( ^{47} C_{4}+ ) ( sum_{j=1}^{5} 52-j C_{3} ) is ( mathbf{A} .^{51} C_{4} ) B. ( ^{52} C_{4} ) c. ( ^{52} C_{3} ) D. ( ^{53} C_{4} ) | 11 |

472 | A question paper has 5 questions. Each question has an alternative. The number of ways in which a student can attempt at least one question is? ( mathbf{A} cdot 2^{5}-1 ) B. ( 3^{5}-1 ) ( c cdot 3^{4}-1 ) D. None of these | 11 |

473 | Six X’s have to be placed in the squares of figure below in such a way that each row contains at least one X. In how many different ways can this be done. (1978) | 11 |

474 | Find the sum of ( left(^{100} C_{0}+^{100} C_{2}+^{100} C_{6}+dots . .right) ) ( mathbf{A} cdot 2^{9} ) в. -2 ( c cdot 2^{5} ) D. ( -2^{5} ) | 11 |

475 | There are six friends of Saurav. In how many ways he can invite one or more friends to take dinner? | 11 |

476 | There are 10 points in a plane of which 4 are collinear. How many different straight lines can be drawn byjoining these points. | 11 |

477 | In how many ways four members be selected from a group of eleven members? This question has multiple correct options | 11 |

478 | There are 720 permutations of the digits ( 1,2,3,4,5,6 . ) suppose these permutations are arranged from smallest to largest numerical of values, beginning from 123456 and ending with ( 654321 . ) (a)what number falls on ( 124^{t h} ) position? (b) what is the position of 321546? | 11 |

479 | If the different permutations of all the letter of the words ( boldsymbol{E} boldsymbol{X} boldsymbol{A} boldsymbol{M} boldsymbol{I} boldsymbol{N} boldsymbol{A} boldsymbol{T} boldsymbol{I} boldsymbol{O} boldsymbol{N} ) are listed as in a dictionary, how many words are there in this list before the first word starting with ( boldsymbol{E} ) ? | 11 |

480 | In how many different orders can five boys stand on a line? A . 40 B. 50 ( c cdot 80 ) D. 120 | 11 |

481 | UUTUULONPUMULIOI IU www 18. These are 10 points in a plane, out of these 6 are collinear, if N is the number of triangles formed by joining these points. then: [2012] (a) ns 100 (b) 100<ns 140 (c) 140<n190 | 11 |

482 | How many strings of letters can possibly by formed using the above rules such that the third letter of the string is e? ( A cdot 8 ) B. 9 ( c cdot 10 ) D. 1 | 11 |

483 | 10. If the LCM of p, q is 24s2, where r, s, t are prime numbers and p, q are the positive integers then the number of ordered pair (p, q) is (2006 – 3M, -1) (a) 252 (b) 254 (@) 225 (d) 224 | 11 |

484 | 7. The number of ways in which 6 men and 5 women can dine at a round table ifno two women are to sit together is given by [2003] (a) 7!* 5! (b) 6! *5! (C) 30 (d) 5! x 4! | 11 |

485 | The value of ( ^{47} C_{4}+sum_{j=1}^{5}(52-j) C_{3} ) is A. ( 47_{5} ) B. ( ^{52} C_{5} ) c. ( ^{52} C_{4} ) D. ( ^{52} C_{3} ) | 11 |

486 | Evaluate: ( ^{71} C_{71} ) | 11 |

487 | 14. Seven white balls and three black balls are randomly placed in a row. The probability that no two black balls are placed adjacently equals (1998 – 2 Marks) (a) 1/2 (b) 7/15 (c) 2/15 (d) 1/3 | 11 |

488 | How many different signals can be given by using any number of flags from six flags of different colors? | 11 |

489 | If ( P_{n} ) denotes the product of all the coefficients in the expansion of ( (1+x)^{n} ) and ( 9 ! P_{(n+1)}=10^{9} P_{n} . ) Then ( boldsymbol{n}= ) A . 10 B. 9 c. 19 D. 11 | 11 |

490 | The number of permutations by taking all letters and keeping the vowels of the word COMBINE in the odd places is ( mathbf{A} cdot 96 ) в. 144 ( c cdot 512 ) D. 576 | 11 |

491 | For a game in which a every pair play with every other pair, 6 men are available. find the number of games which can be played. | 11 |

492 | The number of two-digit numbers which are of the form ( x y ) with ( y<x ) are given by A . 60 B. 55 c. 50 D. 45 | 11 |

493 | The number of such numbers which are divisible by two and five (all digits are not different) is A . 125 B. 76 ( c cdot 65 ) D. 100 | 11 |

494 | The number of ways in which four letters can be selected from the word DEGREE’ is ( A cdot 7 ) B. 6 c. ( frac{6 !}{3 !} ) D. None of these | 11 |

495 | If the letters of the word “VARUN” are written in all possible ways and then are arranged as in a dictionary, then the rank of the word VARUN is? A . 98 B. 99 ( c cdot 100 ) D. 101 | 11 |

496 | A lady gives a dinner party for six guests. The number of ways in which they may be selected from ten friends, if two of the friends will not attend the party together, is? A .112 в. 140 ( c cdot 164 ) D. None of these | 11 |

497 | There are 8 buses running from Kota to Jaipur and 10 buses running from Jaipur to Delhi. In how many ways a person can travel from Kota to Delhi via Jaipur by bus? | 11 |

498 | The number of permutations that can be made out of the letters of the word “MATHEMATICS” When all vowels come together is: A ( cdot frac{8 ! 4 !}{2 !} ) в. ( frac{8 ! 4 !}{(2 !)^{2}} ) c. ( frac{7 ! 4 !}{2 !} ) D. 7!4! | 11 |

499 | How many 3 letter code can be formed by using the five vowels without repetitions? | 11 |

500 | There are 3 books on mathematics 4 on physics and 5 on english.How many different collections can be made such that each collections consists of : Atleast one book of english | 11 |

501 | In a test there were ( n ) questions. In the test ( 2^{n-i} ) students gave wrong answers to at least ( i ) questions ( i=1,2,3 dots . n . ) If the total number of wrong answers given is ( 2047, ) then ( n ) is A . 12 B. 11 c. 10 D. 13 | 11 |

502 | Find the number of arrangements ofthe letters of the word INDEPENDENCE.In how many of | 11 |

503 | Prove that: ( ^{n} C_{r}+^{n} C_{r-1}=^{n+1} C_{r} ) | 11 |

504 | If ( ^{n} C_{12}=^{n} C_{8}, ) find ( ^{n} C_{17},^{22} C_{11} ) | 11 |

505 | How many numbers between 5000 and 10,000 can be formed using the digits ( mathbf{0}, mathbf{1}, mathbf{2}, mathbf{3}, mathbf{4}, mathbf{5}, mathbf{6}, mathbf{7}, mathbf{8}, mathbf{9} ) each digit appearing not more than once in each number ( A cdot 5 times^{8} P_{3} ) В. ( 5 times^{8} C_{3} ) c. ( 5 ! times^{8} P_{3} ) D. ( 5 ! times^{8} C_{3} ) | 11 |

506 | ( mathrm{f} sum_{r=1}^{10} r(r-1)^{10} C_{r}=k .2^{9}, ) then ( k ) is equal to A . 10 B. 45 ( c .90 ) D. 100 | 11 |

507 | Topic-wise Solved Papers Five balls of different colou of different size. Each box can different ways can we place fferent colours are to be placed in there boxes bach box can hold all five. In how many can we place the balls so that no box remains empty ? (1981 – 4 Marks) thot netwo | 11 |

508 | Evaluate ( frac{n !}{(r !) times(n-r) !}, ) when ( n=15 ) and ( r=12 ) | 11 |

509 | Prove that ( sum_{r=0}^{n} 3^{r n} mathbf{C}_{r}=4^{n} ) | 11 |

510 | Ramesh number of ways in which the letters of the word RAMESH can be placed in the squares of the given figure so that no row remains empty, is A . 17280 B. 18720 c. 15840 D. 14400 | 11 |

511 | 5 Indian ( & 5 ) American couples meet at a party & shake hands. If no wife shakes hands with her own husband & no Indian wife shakes hands with male, then the number of hand shakes that takes place in the party is : A . 95 в. 110 ( c .135 ) D. 150 | 11 |

512 | 6. How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even positions ? (2000S) (2) 16 (6) 36 (c) 60 (d) 180 0 | 11 |

513 | 12. The number of seven digit i ntegers, with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only, is (2009) (a) 55 (6) 66 (c) T (d) 88 | 11 |

514 | In a conference there are 11 mechanical engineers and 7 metallurgical engineers. In how many ways can they be seated in a row such that all the metallurgical engineers do not sit together? A. ( 18 !-(12 ! times 7 !) ) B. ( ^{18} P_{4}-2 ! ) ( mathbf{c} cdot^{18} P_{4} times 11 ) D. ( 18 !-11 ! ) | 11 |

515 | There are 6 red, 6 brown, 6 yellow, and 6 gray scarves packaged in 24 identical, unmarked boxes, 1 scarf per box. What is the least number of boxes that must be selected in order to be sure that among the boxes selected 3 or more contain scarves of the same color? A . 3 B. 6 ( c .7 ) D. ( E ) | 11 |

516 | The number of words that can be formed out of the letters of the word “ARTICLE” so that the vowels occupy even places is A .574 B. 36 ( c .754 ) D. 144 | 11 |

517 | When we realize a specific implementation of a pancake algorithm, every move when we find the greatest of the sized array and flipping can be modeled through A. Combinations B. Exponential functions c. Logarithmic functions D. Permutations | 11 |

518 | Let ( x . y . z=105 ) where ( x, y, z in N . ) Then number of ordered triplets ( (x, y, z) ) satisfying the given equation is A . 15 B. 27 ( c cdot 6 ) D. 33 | 11 |

519 | A student is allowed to select at most ( n ) books from a collection of ( (2 n+1) ) books. If the total number of ways in which he can select one book is 63 then the value of ( n ) is A .2 B. 3 ( c cdot 4 ) D. None of these | 11 |

520 | How many 3 -digit even numbers can be formed from the digits 1,2,3,4,5,6 if the digits can be repeated? A ( cdot 108 ) B. 98 c. 72 D. 112 | 11 |

521 | How many different products can be obtained by multiplying two or more of the numbers 3,5,7,11 (without repetition)? | 11 |

522 | Find the quotient of ( 100^{100}+50^{50} ) | 11 |

523 | Find the sum of all four digit numbers that can be formed using the digits ( mathbf{1}, mathbf{3}, mathbf{5}, mathbf{7}, mathbf{9} ) | 11 |

524 | In how many different ways can four books ( A, B, C ) and ( D ) be arranged one above another in a vertical order such that the books ( A ) and ( B ) are never in continuous position? ( A cdot 9 ) B. 12 c. 14 D. 79 | 11 |

525 | If ( ^{15} C_{r}:^{15} C_{r-1}=11: 5 ) find ( r ) | 11 |

526 | Let S = {1,2,3…..9). For k=1.2. …, 5. let N be the number of subsets of S. each containing five elements out of which exactly k are odd. Then N +N,+Nz+NA+Ns = (Jee Adv. 2017) (a) 210 (b) 252 (c) 125 (d) 126 | 11 |

527 | Consider the word ‘PERMUTATION’ Out of all the permutations how many words start with the letter ( M ? ) A. 10 в. ( frac{10 !}{2 !} ) c. 11 D. ( 10 ! times 2 ! ) | 11 |

528 | How many different signals can be made by hoisting 6 differently coloured flags one above the other, when any number of them may be hoisted at once? A . 1956 B . 1955 ( c cdot 1900 ) D. 1901 | 11 |

529 | Let ( y ) be an element of the ( operatorname{set} A= ) {1,2,3,5,6,10,15,30} and ( x_{1}, x_{2}, x_{3} ) be integers such that ( x_{1} x_{2} x_{3}=y, ) then the number of positive integral solutions of ( boldsymbol{x}_{1} boldsymbol{x}_{2} boldsymbol{x}_{3}=boldsymbol{y} ) is ( mathbf{A} cdot 64 ) B . 27 c. 81 D. None of these | 11 |

530 | ( p ) is a prime number and ( n<p<2 n . ) If ( N=^{2 n} C_{n}, ) then A. ( p ) divides ( N ) completely B . ( p^{2} ) divides ( N ) completely c. ( p ) cannot divide ( N ) D. none of these | 11 |

531 | ( f^{35} C_{n+7}=^{35} C_{4 n-2}, ) then the value of ( n ) is ( A cdot 3 ) B. 4 ( c .5 ) D. 6 | 11 |

532 | In how many ways can 6 persons stand in a queue? | 11 |

533 | The number of permutations or the letters of the word EXAMINATION taken 4 at a time is A . 136 в. 2454 ( c .2266 ) D. None of these | 11 |

534 | For ( n ) being natural number, if ( ^{2 n} C_{r}= ) ( ^{2 n} C_{r+2}, ) find ( r ) A ( . n ) B. ( n-1 ) ( mathbf{c} cdot n-2 ) D. ( n-3 ) | 11 |

535 | A box contains two white balls, three black balls and four red balls. No. of ways can three balls be drawn from the box if at least one black ball is to be included in the draw is A . 129 B. 84 c. 64 D. None | 11 |

536 | Which of the following option is the correct combination? A. ( I, i v, S ) в. ( I I I, i i, R ) c. ( I V, I, P ) D. ( I I I, ) iii, ( Q ) | 11 |

537 | How many 3 digit numbers can be formed using the digits 0,1,2,3,4,5,6,7,8 where digits may be repeated? A . 900 в. 980 c. 800 D. 250 | 11 |

538 | Numbers greater than 1000 but not greater than 4000 which can be formed with the digits 0,1,2,3,4 (repetiion of digits is allowed) are A . 350 в. 374 c. 450 D. 575 | 11 |

539 | ( sum_{r=0}^{n} r^{2} .^{n} C_{r} p^{r} q^{n-r}, ) where ( p+q=1 ) is simplified to: A ( cdot n p q+n^{2} p^{2} ) B . ( n^{2} p^{2} q^{2}+n p ) c. ( n p(p+q) ) D. ( frac{p(q+1)}{2} ) | 11 |

540 | The number of ways in which the letters of the word ( A R R A N G E ) can be permuted such that the ( R ) ‘s occur together is A. ( frac{7 !}{2 ! 2 ! ! !} ) в. ( frac{7 !}{2 !} ) c. ( frac{6 !}{2 !} ) D. ( 5 ! times 2 ! ) | 11 |

541 | The number of possible arrangements of letters of the word ( R E V I S I O N ) such that there are exactly two vowels between ( E ) and ( V ) is A. 4032 в. 480 ( c .600 ) D. 720 | 11 |

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