Permutations And Combinations Questions

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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.

Permutations And Combinations Questions

List of permutations and combinations Questions

Question NoQuestionsClass
1Two men enter a railway compartment
having 6 seats unoccupied. In how
many ways can they are seated?		11
2Evaluate the following:
( n+1 C_{n} )		11
3Evaluate
( ^{4} boldsymbol{C}_{3}+^{4} boldsymbol{C}_{4}=? )		11
4The 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
5Determine ( n ) if ( ^{2 n} C_{3}:^{n} C_{3}=11: 1 )11
6How many 3 digit even numbers can be formed using the digits ( 3,5,7,8,9, ) if the digits are not repeated?11
7Find 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
8A 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
9In how many of the permutations of ( n ) things taken ( r ) at a time will 5 things (i)
always occur,
(ii) never occur?		11
10Re. 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
11A 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
12How 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
13The 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
14How many zeros in ( 100 ! ? )11
15A 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
17The 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
18Given 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
19If ( 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
20If ( ^{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
21There 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
22Find the value of ( ^{10} boldsymbol{C}_{5}+mathbf{2} .left(^{10} boldsymbol{C}_{4}right)+^{10} boldsymbol{C}_{3} )11
23How 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
24How many 2 – digit numbers can be formed from the digits ( {1,2,3,4, )
5} without repetition and with repetition?		11
25A 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
26If 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
27How 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
28Find 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
31There 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
32Find the number of arrangements of the letters of the word INDEPENDENCE. In
how many of these arrangements.
Do the vowels never occur together?		11
33Find 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
34A 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
35Prove 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
37Number 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
38There 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
39In 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
40Find the unit digit of ( 2 times 3 times 4 times ldots times )
( mathbf{9 9} )		11
41How many permutations of 4 letters can
be made out of the letters of the word
examination?		11
42How many number greater than a million can be formed with the digits
( 2,3,0,7,7,3,7, ? )		11
43How 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
44There 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
45In 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
46If ( ^{8} C_{r}-^{7} C_{3}=^{7} C_{2}, ) find ( r )11
47Using 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
48If ( 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
49If ( 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
50The value of ( ^{n} C_{1} ) is11
51A 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
52The 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
5313. 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
54A 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
55The value of ( ^{n} C_{n} ) is
( A cdot n )
B.
c. 1
D. ( n ! )		11
56In 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
57Solve the equation ( 3^{x+1} C_{2}+P_{2}^{1} . x= )
( 4^{x} P_{2}, x in N . ) Find ( x ? )		11
58There 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
598.
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
60Find 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
61Twenty 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
62The 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
63Number 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
64The 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
66If
( 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
67How 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
68The 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
69Two 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
70The 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
71Different 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
72If ( 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
73How 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
74The 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
75The 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
76Total 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
77POWIU POSILICI
11.
Prove that
(1989 – 5 Marks)
Co – 22 G + 32 C2 – ……. + (-1)” (n + 1)2 Cn =0,
n>2, where C, = “Cr.		11
78In 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
79There 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
80How 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
82Each 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
83How 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
8417. 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
85A 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
8624.
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
87If ( 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
88Robert 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
89There 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
90A 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
91Find 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
92A 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
93An 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
94A man has 7 trousers and 10 shirts How
many different outfits can he wear?		11
95The 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
96In 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
97Verify that: ( ^{10} C_{4}+^{10} C_{3}=^{11} C_{4} )11
98Number 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
99A 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
100In 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
101How 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
103A regular polygon has 20 sides. How many triangles can be drawn by using the vertices but not using the sides?11
104The 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
105In how many ways can 3 people be seated in a row containing 7 seats?11
106Find 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
107The 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
108There 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
109In 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
110Anagrams 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
111What 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
112The 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
114A 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
115f ( ^{24} C_{x}=^{24} C_{2 x+3}, ) find ( x )11
116The 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
117There 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
118How many different signals can be made by 5 flags from 8 flags of different
colours?		11
119Find 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
120Find 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
121In how many ways 5 different balls can
be distributed into 3 boxes so that no
box remains empty?		11
122If 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
123The 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
124The 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
125The 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
126The 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
127There 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
128Assertion
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
129The 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
13012. 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
131Based 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
132A 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
133We 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
134The 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
135How 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
136From 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
137How 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
138If ( 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
139There 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
140Find the numerically greatest term in the expansion ( (3-5 x)^{15} ) when ( x=frac{1}{5} )11
141Consider 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
142UW.
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
143Find 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
144f
11. The value of 5° ca + § 56-5C is
(a) SSC (b) 55. () 56C;
[2005
(d) 56cm
r=
1		11
145The 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
146The 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
147lului 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
148Find the distinct permutations of the letters of the word MISSISSIPPI?11
149Find 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
150A 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
15120 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
15213.
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
153Eight 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
154When 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
155Out 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
156The 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
157The 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
158At 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
159The 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
161In 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 down11
162Four 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
163How 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
164If ( ^{49} boldsymbol{C}_{3 r-2}=^{49} boldsymbol{C}_{2 r+1} ) find ( ^{prime} boldsymbol{r}^{prime} )11
16524.
· 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
166There 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
A . 10
B. 6
( c cdot 4 )
( D )

11
167golfn-Ic=(K²3) “Erti thenke
la -2, -1/5) [ e c I-V3 31 (4827		11
168The 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
170How 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
17119. 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
172There 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
173Seven 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
174f ( ^{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
175The 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
176If ( ^{28} C_{2 r}:^{24} C_{2 r-4}=225: 11, ) find ( r )11
177The value of ( ^{6} C_{4} ) is
( mathbf{A} cdot mathbf{6} )
B. 9
c. 15
D. 240		11
178The 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
17926. 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
180if ( ^{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
181The 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
182In 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
183Dondue.
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
184Find 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
185There 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
186The 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
188Determine ( n ) if ( ^{2 n} C_{2}:^{n} C_{2}=12: 1 )11
189Assertion
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
190In 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
19121. 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
19222. 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
193How 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
194The 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
195Prove that at any time, the total number of persons on the earth who shake hands an odd number of times is even11
196Choose 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
197Three 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
198The number of combination of 16
things, 8 of which are alike and the rest
different, taken 8 at a time is
( ? )		11
199If ( 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
200Find number of negative integral
solution of equation ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=-mathbf{1 2} )		11
201The 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
202Find 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
204How 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
205If ( ^{n} C_{10}=^{n} C_{15}, ) find ( ^{27} C_{n} )11
206Prove that ( frac{^{n} C_{r}}{n-1}=frac{n}{r} ) where ( 1 leq )
( boldsymbol{r} leq boldsymbol{n} )		11
207If 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
208Evaluate the following:
( ^{35} boldsymbol{C}_{35} )		11
2097 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
210The value of ( ^{19} C_{18}+^{19} C_{17} )
( mathbf{A} cdot 1200 )
B. 2000
( mathbf{C} cdot 190 )
D. None of these		11
211How 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
212Find the sum of all ( 4- ) digit numbers
that can be formed using the digits 0,2,4,7,8 without repetition.		11
213Ten 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
214If ( ^{n} C_{6}:^{n-3} C_{3}=33: 4, ) find ( n )11
215Consider 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
216The 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
217If 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
218Find 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
219How many words can be formed with the
letters of FAILURE when all the vowels
should come together?		11
220The 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
222In how many ways we can select 3
letters of the word PROPORTION?		11
223How 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
224How many 4 digit numbers are there
which contain not more than 2 different
digits?		11
225The number of ways to rearrange the
letters of the word CHEESE is
A . 119
в. 240
( c .720 )
D. 6		11
226How many numbers can be formed from
the digits 1,3,5,9 if repetition of digits is not allowed?		11
22723. 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
228Each 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
229The 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
230Find 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
231If ( 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
233How many different numbers of two
digits can be formed with the digits
1,2,3,4,5,6 no digits being repeated?		11
234A 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
235If 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
237A 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
238Find ( n, ) if ( ^{2 n} C_{3}:^{n} C_{2}=52: 3 )11
239Which 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
24016. 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
241In how many ways three different rings
can be worn in four fingers with at most
one in each finger?		11
242A 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
243Find 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
244Ten 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
245A 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
246Mario’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
247Let ( 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
248Assertion
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
2499.
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
250Find the value of ( 4 ! )11
251The 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
2527.
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
253Find the sum of ( C_{0}+3 C_{1}+3^{2} C_{2}+ )
( ldots+3^{n} C_{n} )		11
2545 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
255From 4 officers and 8 jawans, in how
many ways can 6 be chosen such that to include exactly one officer.		11
256The 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
257Given 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
258There 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
259If ( ^{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
260Different 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
261How 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
262A 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
264If ( ^{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
265What 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
266How 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
267The 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
268There 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
269If ( ^{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
270m 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
271A 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
272If 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
273A 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
275We 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
276The 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
27727.
(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
278If 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
279If ( ^{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
280The 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
281The 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
282There 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
28310. 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
284Assertion
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
285If ( ^{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
287Number 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
288How many words can be formed with the
letters of the word ‘PARALLEL’ so that all
L’s do not come together?		11
289Number 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
290In 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
291In 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
29210. The value of
COT® CO-2)*–20.3.) is where
(2005)		11
293In 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
294In 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
295then
equals (1998 – 2 Marks)
n 1
te a.. = –
r=on C
(a) (n-1),
b) nan
1) None of the above
na		11
296The product of five consecutive numbers is always divisible by?
( mathbf{A} cdot 60 )
B. 12
( c cdot 120 )
D. 72		11
297If ( 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
298In how many different ways can the letters of the word I KURUKSHETRA’ be
arranged?		11
299Let ( 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
300If ( ^{12} C_{r+1}=^{12} C_{3 r-5}, ) find ( r )11
301f ( (n+2) !=2550 times n ! ), find the value
of ( n )		11
302There 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
303Let 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
304There 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
305Assertion
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
306There are three coplanar parallel lines. If any p points are taken on each limes, the maximum number of triangles with vertices at these points is11
307Let ( 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
308In 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
309Delegates 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
310For ( 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
311The 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
312The 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
313Assertion 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
3144 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
315Find 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
3169.
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
317A 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
318A 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
319From 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
320A parallelogram is cut by two sets of ( boldsymbol{m} )
lines parallel to its sides. Find the
number of parallelograms then formed.		11
321The 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
322A 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
323The 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
32520. 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
326There 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
327If
( 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
328The 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
329The 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
330Consider the letters of the word
INTERMEDIATE. Number of words formed
by using all the letters of the given
words are		11
331If 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
( mathrm{R} ) and the fourth letter is ( mathrm{E} ), then the
total number of such words, is?
A ( cdot frac{11 !}{(2 !)^{3}} )
B . 59
c. 110
D. 56

11
332If ( ^{n} C_{8}=^{n} C_{27}, ) then what is the value of
( boldsymbol{n} ? )
A . 35
B. 22
c. 28
D. 41		11
333Determine ( 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
335There 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
336In how many ways can 6 women draw
water from 6 wells, if no well remains
used?		11
337The 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
3385.
For 2 <rsn,
+
1
(2000)
For 2575 n. ()+2(„”.) +(",)- zobas,
@ (m) o (:) *** (*)		11
339If ( 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
340If ( 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
341A 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
342The 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
343The 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
344The 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
345Find the number of arrangements of the
letters of the word INDEPENDENCE. In
how many of these arrangements do all
vowels occur together.		11
346If some or all of ( n ) things be taken at a
time, prove that the number of
combinations is ( 2^{n}-1 )		11
347In 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
348The numbers of ways in whi11
349The 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
350The 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
351The 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
353A 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
354There 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
355How 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
356The 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
357The 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
358The total number of ( 9- ) digit numbers of different digits is
A. ( 10(9 !) )
в. ( 8(9 !) )
c. ( 9(9 !) )
D. none of these		11
35911.
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
360The 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
361A 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
362In 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
363A 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
364In 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
365The 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
366A 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
367There 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
368If ( ^{n} C_{3}=^{n} C_{9}, ) then ( ^{n} C_{2}= )
A . 66
B. 132
c. 72
D. 98		11
369Three 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
370How 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
371Number 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
372The 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
373The 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
374The 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
3751.
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
376If ( 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
377Two 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
378Solve the following inequalities. ( C_{m-2}^{13}>C_{m}^{18}, m epsilon N )11
379How 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
380The product of ( r ) consecutive integers is
divisible by ( r ! )
A. True
B. False		11
381Suppose 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
382Two 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
383Assertion
( 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
384The 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
385State 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
387a) 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
388The number of integer solutions ( (x, y, z) ) of ( x y z=18 ) is
A . 48
B. 64
c. 72
D. 81		11
389A 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
390The 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
3913.
The value of the expression 47
Cz is equal to
(1982 – 2 Marks)
(b) 52C,
(d) none of these
(c) 526		11
392There 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
393The 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
394A 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
395Let ( 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
396A 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
397How 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
398A 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
399If the number of selections of 3
difference letters from the word
( boldsymbol{S U M} boldsymbol{A} boldsymbol{N} )		11
400Given: ( frac{20 !}{18 !}=380 )
A. True
B. False
c. Either
D. Neither		11
401The 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
402If 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
403If ( 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
404How many total distinct terms are there
in the expansion of ( (x+y+z+t)^{10} )		11
405Six 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
406State whether the statement is
true/false.

An die is tossed twice. Find the
probability of getting 4,5 or 6 on the
first toss and 1,2,3 or 4 on the second ( operatorname{toss} ) is ( frac{1}{3} )
A. True
B. False

11
407The 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
408In 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
409Three 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
410Given 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
411A 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
412Find 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
413For 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
414How 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
415Find the value of ( frac{mathbf{6} !}{mathbf{3} !} )11
416Number 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
417There 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
418Arrange 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
419Three 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
420A 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
421A 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
422A 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
423If ( ^{18} boldsymbol{C}_{boldsymbol{r}}=^{18} boldsymbol{C}_{boldsymbol{r}+mathbf{2}}, ) find ( ^{r} boldsymbol{C}_{boldsymbol{5}} )11
424In 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
4253.
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
426In 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
427The 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
428A 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
429Find ( r ) if ( ^{15} C_{3 r}=^{15} C_{r+3} )11
430The 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
431If 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
432The 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
433Number 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
434Find 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
4358064 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
436The exponent of 5 in ( ^{120} C_{60}, ) is
( mathbf{A} cdot mathbf{1} )
B.
( c cdot 2 )
D. 3		11
437The 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
43811. 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
439The 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
440Prove that
( n_{r+1}+2^{n} C_{r}+^{n} C_{r-1}= )		11
44121.
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
442A 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
443The 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
444For ( mathbf{0} leq boldsymbol{r} leq boldsymbol{n},^{boldsymbol{n}} boldsymbol{C}_{boldsymbol{r}}= )11
445Sabnam has 2 school bags, 3 tiffin
boxes and 2 water bottles.show in how
many ways can she carry these item		11
446How many words can be formed from the word “BHARAT”
A . 360
B. 180
( c cdot 90 )
D. 45		11
447On 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
448Number 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
449How 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
450The 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
4511.
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
452Evaluate:
( 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
453The exponent of 18 in ( 200 !, ) is
( mathbf{A} cdot 24 )
B. 46
c. 47
D. 48		11
454Given 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
455Number 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
456The 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
457Eight 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
458A 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
459The 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
461How 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
462The sum of integers from 1 to 100 that are divisible by 2 or
is
[2002]
(a) 3000 (b). 3050 (c) 3600 (d) 3250		11
463Four 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
464How 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
465In 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
466The 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
467How 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
A . 16
B. 36
c. 60
D. 180

11
468In how many ways we can select 5 cards
from a deck of 52 cards, if each selection must include atleast one king		11
469Ten 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
470A 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
471The 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
472A 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
473Six 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
474Find 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
475There are six friends of Saurav. In how
many ways he can invite one or more friends to take dinner?		11
476There are 10 points in a plane of which 4 are collinear. How many different straight lines can be drawn byjoining these points.11
477In how many ways four members be
selected from a group of eleven
members?

This question has multiple correct options
A . ( ^{11} C_{5} )
в. ( ^{11} C_{7} )
( c cdot^{11} C_{4} )
D. ( ^{11} C_{6} )

11
478There 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
479If 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
480In how many different orders can five boys stand on a line?
A . 40
B. 50
( c cdot 80 )
D. 120		11
481UUTUULONPUMULIOI 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
482How 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
48310.
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
4847.
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
485The 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
486Evaluate: ( ^{71} C_{71} )11
48714. 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
488How many different signals can be given by using any number of flags from six flags of different colors?11
489If ( 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
490The 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
491For 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
492The 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
493The 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
494The 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
495If 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
496A 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
497There 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
498The 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
499How many 3 letter code can be formed
by using the five vowels without repetitions?		11
500There 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
501In 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
502Find the number of arrangements ofthe letters of the word

INDEPENDENCE.In how many of
these arrangements so the words start
with ( boldsymbol{P} )

11
503Prove that:
( ^{n} C_{r}+^{n} C_{r-1}=^{n+1} C_{r} )		11
504If ( ^{n} C_{12}=^{n} C_{8}, ) find ( ^{n} C_{17},^{22} C_{11} )11
505How 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
507Topic-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
508Evaluate ( frac{n !}{(r !) times(n-r) !}, ) when ( n=15 )
and ( r=12 )		11
509Prove that ( sum_{r=0}^{n} 3^{r n} mathbf{C}_{r}=4^{n} )11
510Ramesh 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
5115 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
5126.
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
51312.
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
514In 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
515There 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
516The 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
517When 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
518Let ( 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
519A 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
520How 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
521How many different products can be obtained by multiplying two or more of the numbers 3,5,7,11 (without repetition)?11
522Find the quotient of ( 100^{100}+50^{50} )11
523Find 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
524In 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
525If ( ^{15} C_{r}:^{15} C_{r-1}=11: 5 ) find ( r )11
526Let 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
527Consider 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
528How 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
529Let ( 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
532In how many ways can 6 persons stand in a queue?11
533The 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
534For ( 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
535A 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
536Which 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
537How 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
538Numbers 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
540The 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
541The 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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