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Chapter 5 – Permutations and
Combinations

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Question 1
Find x if 1/9! + 1/10! = x/11!
(a) 119

(b) 120

(c) 121

(d) None

CA NISHANT KUMAR 8

Question 1

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Question 2

( n + 1)! − n! = ?
(a) n
...
n

(d) None

CA NISHANT KUMAR 10

Question 3
The number of ways the letters of the word ‘COMPUTER’ can be rearranged is:
(a) 40,320

(b) 40,319

(c) 40,318

(d) None

CA NISHANT KUMAR 11

Question 4
If 18 Cr = 18Cr + 2 , the value of r C5 is:
(a) 55

(b) 50

(c) 56

(d) None

CA NISHANT KUMAR 12

Question 5
In n Pr , the restriction is:
(a) n > r

(b) n ≥ r

(c) n ≤ r

(d) None

CA NISHANT KUMAR 13

Question 6
In n Pr = n ( n − 1)( n − 2 )
...

(a) 20

(b) 21

(c) 22

(d) None

CA NISHANT KUMAR 15

Question 8
In a group of boys the number of arrangement of 4 boys is 12 times the number of
arrangements of 2 boys
...
Find the number of ways of forming
such a committee
...
In how many different ways can the prizes be awarded?
(a) 1,716

(b) 1,720

(c) 1,270

(d) None

CA NISHANT KUMAR 19

Question 12
In how many different ways can 3 students be associated with 4 chartered accountants,
assuming that each chartered accountant can take at most one student?
(a) 24

(b) 25

(c) 30

(d) None

CA NISHANT KUMAR 20

Question 13
When Dr
...
If he
can see only one patient at a time, find the number of ways, he can schedule his patients
(a) if they all want their turn, and (b) if 3 leave in disgust before Dr
...

(a) 12 P12 , 12 P9

(b) 12!, 12 P9

(c) Both

(d) None

CA NISHANT KUMAR 21

Question 14
Mr
...
Y enter into a railway compartment having six vacant seats
...
The number of arrangements is:
(a) 9 8

(b) 10

(c) 8 9

(d) None

CA NISHANT KUMAR 24

Question 17
n articles are arranged in such a way that 2 particular articles never come together
...
The number of such is:
(a) 72

(b) 27

(c) 70

(d) None

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Question 19
The number of ways the letters of the word ‘TRIANGLE’ to be arranged so that the word
‘angle’ will be always present is:
(a) 20

(b) 60

(c) 24

(d) 32

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Question 20
If the letters word ‘DAUGHTER’ are to be arranged so that vowels occupy the odd
places, then number of different words are:
(a) 2,880

(b) 676

(c) 625

(d) 576

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Question 21
How many arrangements can be made out of the letters of the word ‘DRAUGHT’, the
vowels never beings separated?
(a) 1,440

(b) 676

(c) 625

(d) 576

CA NISHANT KUMAR 29

Question 22
There are 6 books on Economics, 3 on Mathematics and 2 on Accountancy
...
In
how many ways can they be seated if (i) all the sisters sit together, (ii) no two sisters sit
together?
(a) 720; 1,440

(b) 1,440; 720

(c) 840; 720

(d) None

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Question 23

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Question 24
Six boys and five girls are to be seated for a photograph in a row such that no two girls
sit together and no two boys sit together
...

(a) 72,000

(b) 14,440

(c) 86,400

(d) None

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Question 25
The number of arrangements of 10 different things taken 4 at a time in which one
particular thing always occurs is:
(a) 2015

(b) 2016

(c) 2014

(d) None

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Question 26
The number of permutations of 10 different things taken 4 at a time in which one
particular thing never occurs is:
(a) 3,020

(b) 3,025

(c) 3,024

(d) None

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Question 27
The number of numbers lying between 10 and 1000 can be formed with the digits 2, 3,
4, 0, 8, 9 is:
(a) 124

(b) 120

(c) 125

(d) None

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Question 28
The total number of 9 digit numbers of different digits is:
(a) 10 9

(b) 8 9

(c) 9 9

(d) None

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Question 29
There are 5 speakers A, B, C, D and E
...
The number of ways in which a
person can go from Calcutta to Delhi and return by a different train is:
(a) 99

(b) 90

(c) 80

(d) None

CA NISHANT KUMAR 39

Question 31
Number ways of painting of a face of a cube by 6 colours is:
(a) 36

(b) 6

(c) 24

(d) 20

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Question 32
How many six-digit telephone numbers can be formed by using 10 distinct digits?
(a) 108

(b) 610

(c) 10 C9

(d) 10 P6

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Question 33
The number of ways in which 8 sweats of different sizes can be distributed among 8
persons of different ages so that the largest sweat always goes to be younger assuming
that each one of then gets a sweat is:
(a) 8

(b) 5040

(c) 5039

(d) None

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Question 34
The number of arrangements in which the letters of the word ‘MONDAY’ be arranged
so that the words thus formed begin with M and do not end with N is:
(a) 720

(b) 120

(c) 96

(d) None

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Question 35
How many numbers of seven-digit numbers which can be formed from the digits 3, 4, 5,
6, 7, 8, 9, no digit being repeated are not divisible by 5?
(a) 4320

(b) 4690

(c) 3900

(d) 3890

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Question 35

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Question 36
Eight guests have to be seated 4 on each side of a long rectangular table
...
The number of ways
in which the sitting arrangements can be made is:
(a) 1732

(b) 1728

(c) 1730

(d) 1278

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Question 36

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Question 37
The number of even numbers greater than 300 can be formed with the digits 1, 2, 3, 4, 5
without repetition is:
(a) 110

(b) 112

(c) 111

(d) None

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Question 37

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Question 37

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Question 38
The letters of the words ‘CALCUTTA’ and ‘AMERICA’ are arranged in all possible
ways
...
The number of guests in the party is:
(a) 11

(b) 12

(c) 13

(d) 14

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Question 41
There are 12 points in a plane of which 5 are collinear
...
In how many ways can he invite 7 guests
such that 5 of them are relatives?
(a) 420

(b) 446

(c) 336

(d) None

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Question 44
The number of parallelograms that can be formed from a set of four parallel lines
intersecting another set of three parallel lines is:
(a) 6

(b) 18

(c) 12

(d) 9

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Question 45
The Supreme Court has given a 6 to 3 decision upholding a lower court; the number of
ways it can give a majority decision reversing the lower court is:
(a) 256

(b) 276

(c) 245

(d) 226

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Question 46
Five bulbs of which three are defective are to be tried in two bulb points in a dark room
...
The number of committees
such that each committee includes at least one lady is:
(a) 400

(b) 440

(c) 441

(d) None

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Question 49
There are 7 men and 3 ladies
...
In how many ways can this be done if in the committee, there
must be at least one member from each group and at least 3 Chartered Accountants?
(a) 3,570

(b) 3,750

(c) 7,350

(d) None

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Question 50

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Question 51
A committee of 3 ladies and 4 gents is to be formed out of 8 ladies and 7 gents
...
X
refuses to serve in a committee in which Mr
...
The number of such
committees is:
(a) 1530

(b) 1500

(c) 1520

(d) 1540

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Question 51

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Question 52
A boy has 3 library tickets and 8 books of his interest in the library
...
In
how many ways can he choose the three books to be borrowed?
(a) 41

(b) 51

(c) 61

(d) 71

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Question 52

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Question 53
The ways of selecting 4 letters from the word ‘EXAMINATION’ is
(a) 136

(b) 130

(c) 125

(d) None

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Question 53

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Question 53

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Question 54
The number of 4-digit numbers formed with the digits 1, 1, 2, 2, 3, 4 is:
(a) 100

(b) 101

(c) 201

(d) None

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Question 54

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Question 54

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Question 55
In how many ways can 8 persons be seated at a round table? In how many cases will 2
particular persons sit together?
(a) 5,040; 1,440

(b) 5,040; 720

(c) 5,040; 120

(d) None

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Question 56
The number of ways in which 7 boys sit in a round table so that two particular boys may
sit together is:
(a) 240

(b) 200

(c) 120

(d) None

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Question 57
3 ladies and 3 gents can be seated at a round table so that any two and only two of the
ladies sit together
...
Find the number of ways in which they can be seated
...
8!
(a)

7!
8!
15
(d) 2
...
The number of different
forecasts containing exactly 6 correct results is:
(a) 316

(b) 214

(c) 112

(d) None

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Question 62
Eight chairs are numbered from 1 to 8
...
First, the women choose the chairs from the chairs numbered
1 to 4 and then men select the chairs from the remaining
...

The maximum number of trials that are needed to assigns the keys to the corresponding
locks is:
(a)

( n −1)

C2

(b)

( n +1)

n

C2

(c)

 ( k − 1)
k =2

n

(d)

k
k =2

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Question 63

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Some Standard Results
1
( n − 1)!
...
Number of ways of selecting some or all items from a set of n items –
a
...


1
...
When there are 3 choices for each item: ( 3n − 1)
...
n
; n
Cr +1 n − r
Cr
n − r +1
3
...
If n Cx = nC y , and x  y , then x + y = n
...
If n Px = n Py , and x  y , then x + y = 2n − 1
...
The number of diagonals in a polygon of n sides is

1
n ( n − 3)
...
Division of Items in Groups –
a
...
Equal items in every group – The number of ways to divide n students
n!
into k groups of h students each is given by

...
Unequal items in every group – The number of ways to divide n items into
3 groups → one containing a items, the second containing b items, and
n!
the third containing c items, such that a + b + c = n , is given by

...
Division of Identical Items in Groups – The number of ways to divide n
n!
identical objects into k groups of h items each is given by

...
Number of Factors of a number – Factors of a number N refers to all the numbers
which divide N completely
...
qb
...

Step 2 – Use the formula: Number of factors of N = ( a + 1)( b + 1)( c + 1)
...


The maximum number of points of intersection of n circles will be n ( n − 1)
...
At least one question from each section is to be attempted
...

Therefore, 26 − 1
...
Therefore, 24 − 1
...


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Question 66
There are 12 questions to be answered in Ye

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