4

Permutations
and Combinations
7. In a collage of 300 students, every student reads
Topic 1 General Arrangement 5 newspapers and every newspaper is read by 60
students. The number of newspapers is (1998, 2M)
Objective Questions I (Only one correct option) (a) atleast 30
1. The number of four-digit numbers strictly greater than (b) atmost 20
4321 that can be formed using the digits 0, 1, 2, 3, 4, 5 (c) exactly 25
(repetition of digits is allowed) is (2019 Main, 8 April II) (d) None of the above
(a) 306 (b) 310 8. A five digits number divisible by 3 is to be formed using
(c) 360 (d) 288 the numbers 0, 1 , 2, 3 , 4 and 5, without repetition. The
2. How many 3 × 3 matrices M with entries from {0, 1, 2} total number of ways this can be done, is (1989, 2M)
are there, for which the sum of the diagonal entries of (a) 216 (b) 240
M T M is 5 ? (2017 Adv.) (c) 600 (d) 3125
(a) 198 (b) 162 (c) 126 (d) 135 9. Eight chairs are numbered 1 to 8. Two women and
3. The number of integers greater than 6000 that can be three men wish to occupy one chair each.
formed using the digits 3, 5, 6, 7 and 8 without First the women choose the chairs from amongst the
repetition is (2015 Main) chairs marked 1 to 4 and then the men select the chairs
(a) 216 (b) 192 from amongst the remaining. The number of possible
(c) 120 (d) 72 arrangements is
(a) 6C3 × 4 C2 (b) 4 P2 × 4 P3 (1982, 2M)
4. The number of seven-digit integers, with sum of the
digits equal to 10 and formed by using the digits 1, 2 and (c) 4 C2 + 4 P3 (d) None of these
3 only, is (2009) 10. The different letters of an alphabet are given. Words
(a) 55 (b) 66 (c) 77 (d) 88 with five letters are formed from these given letters.
Then, the number of words which have at least one
5. How many different nine-digit numbers can be formed
letter repeated, is (1980, 2M)
from the number 22 33 55 888 by rearranging its digits
so that the odd digits occupy even positions? (2000, 2M) (a) 69760 (b) 30240
(c) 99748 (d) None
(a) 16 (b) 36
(c) 60 (d) 180
Analytical & Descriptive Question
6. An n-digit number is a positive number with exactly n
digits. Nine hundred distinct n-digit numbers are to be 11. Eighteen guests have to be seated half on each side of a
formed using only the three digits 2,5 and 7. The smallest long table. Four particular guests desire to sit on one
value of n for which this is possible, is (1998, 2M) particular side and three other on the other side.
(a) 6 (b) 7 (c) 8 (d) 9 Determine the number of ways in which the sitting
arrangements can be made. (1991, 4M)
 Permutations and Combinations 77

Match the Columns

Match the conditions/expressions in Column I with statement in Column II.
12. Consider all possible permutations of the letters of the word ENDEANOEL. (2008, 6M)

Column I Column II
A. The number of permutations containing the word p. 5!
ENDEA, is
B. The number of permutations in which the letter E q. 2 × 5!
occurs in the first and the last positions, is
C. The number of permutations in which none of the r. 7 × 5!
letters D, L, N occurs in the last five positions, is
D. The number of permutations in which the letters A, E, s. 21 × 5!
O occur only in odd positions, is

Topic 2 Properties of Combinational and General Selections

Objective Questions I (Only one correct option) 6. If n C 4 , n C 5 and n C 6 are in AP, then n can be
1. There are 3 sections in a question paper and each (2019 Main, 12 Jan II)
section contains 5 questions. A candidate has to answer (a) 9 (b) 11 (c) 14 (d) 12
25
a total of 5 questions, choosing at least one question
from each section. Then the number of ways, in which 7. If ∑{
r = 0
50
Cr ⋅ 50 − r
C 25 − r } = K (50C 25 ),
the candidate can choose the questions, is
[2020 Main, 5 Sep II] then, K is equal to (2019 Main, 10 Jan II)

(a) 3000 (b) 1500 (a) 224 (b) 225 − 1 (c) 225 (d) (25)2
3
(c) 2255 (d) 2250 20 20
Ci − 1  k
2. The number of ways of choosing 10 objects out of 31
8. If ∑ 
i=1
20 20
 = , then k equals
Ci + Ci − 1  21 (2019 Main, 10 Jan I)
objects of which 10 are identical and the remaining 21
are distinct, is (2019 Main, 12 April I) (a) 100 (b) 400 (c) 200 (d) 50
(a) 2 20
−1 (b) 221
(c) 2 20
(d) 2 20
+1 9. A man X has 7 friends, 4 of them are ladies and 3 are
men. His wife Y also has 7 friends, 3 of them are ladies
3. Suppose that 20 pillars of the same height have been
and 4 are men. Assume X and Y have no common
erected along the boundary of a circular stadium. If the
friends. Then, the total number of ways in which X and
top of each pillar has been connected by beams with the
Y together can throw a party inviting
top of all its non-adjacent pillars, then the total number
3 ladies and 3 men, so that 3 friends of each of X and Y
of beams is (2019 Main, 10 April II)
are in this party, is (2017 Main)
(a) 180 (b) 210 (c) 170 (d) 190
(a) 485 (b) 468 (c) 469 (d) 484
4. Some identical balls are arranged in rows to form an
equilateral triangle. The first row consists of one ball,
10. Let S = {1, 2, 3, …… , 9}. For k = 1, 2 , …… 5, let N k be the
the second row consists of two balls and so on. If 99 more number of subsets of S, each containing five elements
identical balls are added to the total number of balls out of which exactly k are odd. Then
used in forming the equilateral triangle, then all these N1 + N 2 + N 3 + N 4 + N 5 = (2017 Adv.)
balls can be arranged in a square whose each side (a) 210 (b) 252 (c) 126 (d) 125
contains exactly 2 balls less than the number of balls 11. A debate club consists of 6 girls and 4 boys. A team of
each side of the triangle contains. Then, the number of 4 members is to be selected from this club including the
balls used to form the equilateral triangle is selection of a captain (from among these 4 members) for
(2019 Main, 9 April II) the team. If the team has to include atmost one boy, the
(a) 262 (b) 190 (c) 225 (d) 157 number of ways of selecting the team is (2016 Adv.)
5. There are m men and two women participating in a (a) 380 (b) 320 (c) 260 (d) 95
chess tournament. Each participant plays two games
with every other participant. If the number of games
12. Let Tn be the number of all possible triangles formed by
played by the men between themselves exceeds the joining vertices of an n-sided regular polygon. If
number of games played between the men and the Tn + 1 − Tn = 10, then the value of n is (2013 Main)
women by 84, then the value of m is (2019 Main, 12 Jan II) (a) 7 (b) 5
(a) 12 (b) 11 (c) 9 (d) 7 (c) 10 (d) 8
78 Permutations and Combinations

13. If r , s, t are prime numbers and p, q are the positive 17. Let n ≥ 2 be an integer. Take n distinct points on a circle
integers such that LCM of p, q is r 2s4 t 2 ,then the number and join each pair of points by a line segment. Colour the
of ordered pairs ( p, q) is (2006, 3M) line segment joining every pair of adjacent points by blue
(a) 252 (b) 254 and the rest by red. If the number of red and blue line
segments are equal, then the value of n is (2014 Adv.)
(c) 225 (d) 224
5
Fill in the Blanks
14. The value of the expression 47 C 4 + ∑
j =1
52− j
C 3 is
18. Let A be a set of n distinct elements. Then, the total
(1980, 2M)
47
(a) C5 number of distinct functions from A to A is…and out of
(b) 52C5 these… are onto functions. (1985, 2M)
(c) 52C4 19. In a certain test, a i students gave wrong answers to at
(d) None of these least i questions, where i = 1, 2, K , k. No student gave
more that k wrong answers. The total number of wrong
Match the Columns answers given is … . (1982, 2M)

15. In a high school, a committee has to be formed from a True/False

group of 6 boys M1 , M 2, M 3, M 4 , M 5, M 6 and 5 girls G1 ,
G2, G 3, G 4 , G 5. 20. The product of any r consecutive natural numbers is
always divisible by r !. (1985, 1M)
(i) Let α1 be the total number of ways in which the
committee can be formed such that the committee Analytical & Descriptive Questions
has 5 members, having exactly 3 boys and 2 girls.
21. A committee of 12 is to be formed from 9 women and 8
(ii) Let α 2 be the total number of ways in which the men. In how many ways this can be done if at least five
committee can be formed such that the committee women have to be included in a committee ? In how
has at least 2 members, and having an equal many of these committees
number of boys and girls. (i) the women are in majority?
(iii) Let α 3 be the total number of ways in which the (ii) the men are in majority? (1994, 4M)
committee can be formed such that the committee
22. A student is allowed to select atmost n books from n
has 5 members, at least 2 of them being girls.
collection of (2n + 1) books. If the total number of ways
(iv) Let α 4 be the total number of ways in which the in which he can select at least one books is 63, find the
committee can be formed such that the committee value of n. (1987, 3M)
has 4 members, having at least 2 girls such that both
M1 and G1 are NOT in the committee together.
23. A box contains two white balls, three black balls and
(2018 Adv.)
four red balls. In how many ways can three balls be
drawn from the box, if at least one black ball is to be
List-I List-II included in the draw ? (1986, 2 12 M)
P. The value of α1 is 1. 136 24. 7 relatives of a man comprises 4 ladies and 3
Q. The value of α 2 is 2. 189 gentlemen, his wife has also 7 relatives ; 3 of them are
ladies and 4 gentlemen. In how many ways can they
R. The value of α 3 is 3. 192 invite a dinner party of 3 ladies and 3 gentlemen so
that there are 3 of man’s relative and 3 of the wife's
S. The value of α 4 is 4. 200
relatives? (1985, 5M)
5. 381 25. m men and n women are to be seated in a row so that no
6. 461 two women sit together. If m > n, then show that the
number of ways in which they can be seated, is
The correct option is m ! (m + 1) !
.
(a) P → 4; Q → 6; R → 2; S → 1 (m − n + 1) ! (1983, 2M)
(b) P → 1; Q → 4; R → 2; S → 3
26. mn squares of equal size are arranged to form a
(c) P → 4; Q → 6; R → 5; S → 2 rectangle of dimension m by n where m and n are
(d) P → 4; Q → 2; R → 3; S → 1 natural numbers. Two squares will be called
‘neighbours’ if they have exactly one common side. A
Integer & Numerical Answer Type Questions natural number is written in each square such that the
number in written any square is the arithmetic mean of
16. Five persons A , B, C , D and E are seated in a circular the numbers written in its neighbouring squares. Show
arrangement. If each of them is given a hat of one of the that this is possible only if all the numbers used are
three colours red, blue and green, then the number of equal. (1982, 5M)
ways of distributing the hats such that the persons
27. If n C r −1 = 36, n C r = 84 and n C r +1 = 126, then find the
seated in adjacent seats get different coloured hats is
……… (2019 Adv.) values of n and r. (1979, 3M)
 Permutations and Combinations 79

Topic 3 Multinomial, Repeated Arrangement and Selection

Objective Question I (Only one correct option) Integer & Numerical Answer Type Questions
1. The number of 6 digits numbers that can be formed 8. An engineer is required to visit a factory for exactly four
using the digits 0, 1, 2,5, 7 and 9 which are divisible by days during the first 15 days of every month and it is
11 and no digit is repeated, is (2019 Main, 10 April I) mandatory that no two visits take place on consecutive
(a) 60 (b) 72 days. Then the number of all possible ways in which
(c) 48 (d) 36 such visits to the factory can be made by the engineer
during 1-15 June 2021 is ……… (2020 Adv.)
2. A committee of 11 members is to be formed from 8 males
and 5 females. If m is the number of ways the committee 9. If the letters of the word ‘MOTHER’ be permuted and
is formed with at least 6 males and n is the number of all the words so formed (with or without meaning) be
ways the committee is formed with atleast 3 females, listed as in dictionary, then the position of the word
‘MOTHER’ is………… (2020 Main, 2 Sep I)
then (2019 Main, 9 April I)
(a) m = n = 68 (b) m + n = 68 10. The number of 5 digit numbers which are divisible by 4,
(c) m = n = 78 (d) n = m − 8 with digits from the set {1, 2, 3, 4, 5} and the repetition of
digits is allowed, is ..................... . (2018 Adv.)
3. Consider three boxes, each containing 10 balls labelled
1, 2, …, 10. Suppose one ball is randomly drawn from 11. Words of length 10 are formed using the letters A, B, C,
each of the boxes. Denote by ni , the label of the ball D, E, F, G, H, I, J. Let x be the number of such words
drawn from the ith box, (i = 1, 2, 3). Then, the number of where no letter is repeated; and let y be the number of
ways in which the balls can be chosen such that such words where exactly one letter is repeated twice
y
n1 < n2 < n3 is (2019 Main, 12 Jan I) and no other letter is repeated. Then, =
9x (2017 Adv.)
(a) 82 (b) 120
(c) 240 (d) 164 12. Let n be the number of ways in which 5 boys and 5 girls
can stand in a queue in such a way that all the girls
4. The number of natural numbers less than 7,000 which
stand consecutively in the queue. Let m be the number
can be formed by using the digits 0, 1, 3, 7, 9 (repitition
of ways in which 5 boys and 5 girls can stand in a queue
of digits allowed) is equal to (2019 Main, 9 Jan II)
in such a way that exactly four girls stand consecutively
(a) 374 (b) 375 m
in the queue. Then, the value of is
(c) 372 (d) 250 n (2015 Adv.)
5. Consider a class of 5 girls and 7 boys. The number of 13. Let n1 < n2 < n3 < n4 < n5 be positive integers such that
different teams consisting of 2 girls and 3 boys that can
n1 + n2 + n3 + n4 + n5 = 20. The number of such distinct
be formed from this class, if there are two specific boys
arrangements (n1 , n2 , n3 , n4 , n5 ) is (2014 Adv.)
A and B, who refuse to be the members of the same
team, is (2019 Main, 9 Jan I) Fill in the Blanks
(a) 350 (b) 500 k(k + 1)
(c) 200 (d) 300 14. Let n and k be positive integers such that n ≥ .
2
6. If all the words (with or without meaning) having five The number of solutions (x1 , x2 ,... , xk ),
letters, formed using the letters of the word SMALL and x1 ≥ 1, x2 ≥ 2, ... , xk ≥ k for all integers satisfying
arranged as in a dictionary, then the position of the x1 + x2 + ... + xk = n is … (1996, 2M)
word SMALL is (2016 Main) 15. Total number of ways in which six ‘+’ and four ‘–’ signs
(a) 46th (b) 59th can be arranged in a line such that no two ‘–’signs occur
(c) 52nd (d) 58th together is… . (1988, 2M)
7. The letters of the word COCHIN are permuted and all
the permutations are arranged in an alphabetical order Analytical & Descriptive Question
as in an English dictionary. The number of words that 16. Five balls of different colours are to be placed in three
appear before the word COCHIN, is (2007, 3M) boxes of different sizes. Each box can hold all five. In
(a) 360 (b) 192 how many different ways can we place the balls so that
(c) 96 (d) 48 no box remains empty? (1981, 4M)
80 Permutations and Combinations

Topic 4 Distribution of Object into Group

Objective Questions I (Only one correct option) 4. The total number of ways in which 5 balls of different
colours can be distributed among 3 persons so that each
1. A group of students comprises of 5 boys and n girls. If
person gets at least one ball, is (2012)
the number of ways, in which a team of 3 students can
randomly be selected from this group such that there is (a) 75 (b) 150 (c) 210 (d) 243
at least one boy and at least one girl in each team, is 5. The number of arrangements of the letters of the word
1750, then n is equal to (2019 Main, 12 April II) BANANA in which the two N’s do not appear adjacently,
(a) 28 (b) 27 is (2002, 1M)
(c) 25 (d) 24 (a) 40 (b) 60 (c) 80 (d) 100
2. Let S be the set of all triangles in the xy-plane, each Analytical & Descriptive Questions
having one vertex at the origin and the other two
n2 !
vertices lie on coordinate axes with integral 6. Using permutation or otherwise, prove that is an
coordinates. If each triangle in S has area 50 sq. units, (n !)n
then the number of elements in the set S is integer, where n is a positive integer. (2004, 2M)
(2019 Main, 9 Jan II)
7. In how many ways can a pack of 52 cards be
(a) 36 (b) 32
(i) divided equally among four players in order
(c) 18 (d) 9
(ii) divided into four groups of 13 cards each
3.. From 6 different novels and 3 different dictionaries, 4 (iii) divided in 4 sets, three of them having 17 cards each
novels and 1 dictionary are to be selected and arranged and the fourth just one card? (1979, 3M)
in a row on a shelf, so that the dictionary is always in
the middle. The number of such arrangements is Integer & Numerical Answer Type Question
(2018 Main)
(a) atleast 1000
8. In a hotel, four rooms are available. Six persons are to
be accommodated in these four rooms in such a way that
(b) less than 500
each of these rooms contains at least one person and at
(c) atleast 500 but less than 750
most two persons. Then the number of all possible ways
(d) atleast 750 but less than 1000
in which this can be done is ……… (2020 Adv.)

Topic 5 Dearrangement and Number of Divisors

Objective Question I (Only one correct option) Fill in the Blank
1. Number of divisors of the form (4n + 2), n ≥ 0 of the 2. There are four balls of different colours and four boxes of
integer 240 is (1998, 2M) colours, same as those of the balls. The number of ways in
(a) 4 (b) 8 which the balls, one each in a box, could be placed such
(c) 10 (d) 3
that a ball does not go to a box of its own colour is.... .
(1992, 2M)

Answers
Topic 1 Topic 3
1. (b) 2. (a) 3. (b) 4. (c) 1. (a) 2. (c) 3. (b) 4. (a)
5. (c) 6. (b) 7. (c) 8. (a) 5. (d) 6. (d) 7. (c) 8. (495)
9. (d) 10. (a) 11. 9
P4 × 9 P3 (11 )! 9. (309) 10. (625) 11. (5) 12. (5)
1
12. ( A → p; B → s; C → q ; D → q ) 13. (7) 14. (2n − k + k − 2 )
2
15. (35 ways)
2
Topic 2 16. (300)
1. (d) 2. (c) 3. (c) 4. (b)
Topic 4
5. (a) 6. (c) 7. (c) 8. (a)
1. (c) 2. (a) 3. (a) 4. (b)
9. (a) 10. (c) 11. (a) 12. (b)
(52 )! (52 )! (52 )!
13. (c) 14. (c) 15. (c) 16. (30) 5. (a) 7. (i) 4
(ii) 4
(iii)
n (13 !) 4 ! (13 !) 3 ! (17 ) 3
17. (5) 18. n , ∑ ( −1 )
n n −r n
Cr (r ) n
19. 2 − 1
n
8. (1080)
r =1

20. (True) 21. 6062, (i) 2702 (ii) 1008 22. n = 3 Topic 5
23. (64) 24. (485) 27. (n = 9 and r = 3 ) 1. (a) 2. (9)