Content marketed & distributed by FaaDoOEngineers.com PERMUTATION & COMBINATION By:- Nishant Gupta
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PERMUTATION & COMBINATION
Permutation :
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-85 Contact: 9953168795, 9268789880
Content marketed & distributed by FaaDoOEngineers.com Each of arrangement which can be made by some or all of a number of things i.e. in this case order in which we put things matters. Like (1,2,3) 123, 132, 213, 231, 312, 321 Combination : Here order doesn’t matter. It means number of ways of choosing r things out of given things
n! n r !
Number of permutations of taking r of n is
n
Pr
While combination is
n
Cr
( n C r is also represented as C r or C ( n,r)
n! n r ! r !
Some important relations n
C r n C r 1
n 1
C r 1
,
C1 + C2 + … + Cn= 2n -1
n n
Cr
C r 1
n 1 r , r
Co + C2 + C4 + …=
n
Cr =
n n 1 C r 1 r
C1 + C3 + C5 + …= 2n-1
Counting formula :-No. of ways of dividing n identical things among r persons, each can receive none or any no. of things is
n r 1
In above if each gets at least one then
n 1
Coefficient of x r
in
C r 1 n r 1
( 1-x )- n is
No. of ways in which n things can be arranged in row is
In circular arrangement
Arranging n objects along a circle when clock & anticlockwise are considered alike is ( like neckless of n beads )
C r 1
C n 1
n! (n – 1) ! ½ (n – 1) !
No. of hand shakes of n people each shakes hand with all others
No. of lines formed by n points of which m are collinear is
n
C2 m C2 1
No. of s in above case is
n
C3 m C3
No. of diagonals in n sided polygon
n
C2 n =
No. of rectangles by two sets of | | lines having n & m lines
n
C 2 m C 2
If 2m things are equally divided between two groups ( Order not important ) then number of ways
C2
n n 3 2
2m! m! m!2!
If 2m things are equally divided between two groups ( Order important ) then number of ways
n
2m! 2! = m! m!2!
2m! m! m!
No. of ways of dividing n + m + r things into groups ( Order not important ) containing n,m,r things is
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-85 Contact: 9953168795, 9268789880
n m r ! m! n! r!
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No. of ways of dividing n + m + r things into persons containing n,m,r things is
If n m different things are equally divided m groups then number of ways
mn ! 1 n!m m!
n m r ! X 3! m! n! r!
( if order of group is not important)
mn ! ( if order of group is important) n!m
Number of de-arrangements ( i.e. number of ways n things can be arranged such that none is at the place assigned to it )
n! (
1 1 1 1 ...... ) 2! 3! 4 n!
Restricted Permutations No. of ways of n different objects taken r at a time in which m objects are always (i) Excluded is
n m
Pr or
n m
(ii) Included
Cr
n m
Pr m X
r
Pm
Number and Sum of divisors of a given number :First write the given number as product of prime numbers. Let it be
P1R .P2S........PnZ
No. of divisors = (R+1) (S+1)………(Z+1) = N ( say ) For number of proper divisors subtract 2 from N
P1R 1 1 P1S1 1 P Z1 1 .......... n P1 1 P2 1 Pn 1
(i) Sum of divisors is
(ii) To get number of odd divisors apply (i) for odd factors (iii) To get number of even divisors
N minus odd divisors
Sum of numbers formed by taking all the given n digits = (sum of digits) x (n-1) ! (111…………… n times). Sum of digits at any of particular places ( unit , ten etc ) formed by taking all the given n digits = (sum of digits) x (n-1) ! No. of possible selections of n + m + r things of which n are all alike & so are m & r is ((n + 1) ( m + 1) (r + 1) – 1 Exponent of any number ( say x ) in y !
y y y x 2 3 ..... , x x
[ . ] is Greatest integer function
No. of ways in which n things of which p, q, r are alike of one, second & third kind is n!/(p! q! r!) No. of ways in which m (one type of different) things & n (another type of different) things can be arranged in a row s.t. all the things of second type come together is n! (m + 1) ! In case of row already having n things there are n + 1 alternate places In case of circle already having n things there are n alternate places
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-85 Contact: 9953168795, 9268789880
Content marketed & distributed by FaaDoOEngineers.com NOTE : Always be careful in ALL ARE NOT TOGETHER & NO TWO ARE TOGETHER
3
3.
14
C7
17i
C6
r 1
(a)
16
C7
(b)
17
C7
(c)
17
C8
(d)
16
C8
(b)
99
5
4.
5.
Assignment 1.
2.
95 C 4 (a)
99
(c)
100
100 j
C3
j1
C5
(d) N/T
C4
If (15 C r 15 C r 1 )( 15 C15r 15 C16r ) (16 C13 ) 2 then r is (a) 3
(b) 2
(c) 4
(d) None.
If a and b are the greatest values of 2n – 1C respectively. Then : r (a) a = 2b
(b) 2a = b
(c) a = b
(d) N/T.
7. 2nC
r
and
C x 1 2
Smallest x satisfying
10
(a) 7
(b) 10
(c) 9 6.
C4
10
C x is
(d) 8
If C r C 3 C 2 then r 8
7
7
(a) 2 or 6
(b) 3 or 5
(c) 3 or 4
(d) 4 or 5
The number of positive integers satisfying the inequality n – 1 Cn – 2 – n + 1 C n – 1 100 is : (a) 9
(b) 5
(c) 8
(d) N/T
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-85 Contact: 9953168795, 9268789880
Content marketed & distributed by FaaDoOEngineers.com 8.
9.
10.
Least value of n such that C(n, 5) + C( n, 6) > C(n +1, 5) (a) 10
(b) 12
(c) 13
(d) 11
Tn is number of s formed using vertices of n sided polygon of. If Tn+1 – Tn = 21 then n is (a) 5
(b) 7
(c) 6
(d) 4
There are 20 pt. in a plane of which no three except 7 are collinear. No. of s there can be formed using these pts. are vertices is (a) (c)
11.
12.
72
C4
13
(a)
32
(c)
22
15.
16.
22
C 3 10 C 3
(d) N/T
In how many ways can a football team of 11 players be selected from 16 player’s so as to exclude 2 particular players 16
C9
18.
19.
20.
C 3 10 C 3
(b)
21.
(b) 264 (d) N/T
22.
No. of 4 digit no. made with 1,2,…..,5 s.t atleast 2 digits identical (a) 45 – 5!
(b) 505
(c) 600
(d) N/T
23.
No. of 5 digit no. when no two consecutive digits are identical (a) 92 x 83
(b) 9 x 84
(c) 95
(d) 8 x 94
A convex polygon of n sides has twice as many diagonals as the number of sides. The value of n is
(c) 4!
(d) N/T
Number of three digit odd numbers using {1,2,3,7,4,5,6 } (c) 120
(d) N/T
C3
(b) 4 C 2 4 C 2 x 4!
(a) 196
C3 7 C3
C3 7 C3
(c) 364
14.
17
There are 22 pt. in a plane of which no three except 10 are collinear. No. of s there can be formed using these pts. are vertices is
(a) 13.
(b)
17.
(a) 4 P2 4 P2
24.
`
(b) 60 (d) 1
There are 4 oranges, 5 apples & 3 mangoes. Number of ways of making selections of fruits (a) 210
(b) 119
(c) 209
(d) 920
8 identical coins are arranged in a row, number of ways these show equal number of heads & tails is (a) 70
(b) 120
(c) 72
(d) 40
How many words can be made from the letters of the word DELHI, if L comes in the middle in every word? (a) 60
(b) 24
(c) 12
(d) 6
Ways of arranging 5 players to throw cricket ball so that youngest may not throw first is (a) 119
(b) 95
(c) 24
(d) 96
Number of 5 digit number consisting of different odd digits & not divisible by 5 (a) 2500
(b) 24
(c) 96
(d) 120
Number of six digit numbers 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) 288
(b) 720
(c) 144
(d) 72
No. of ways in which 5 letters can be posted in 10 letter boxes
(a) 5
(b) 6
(a) 510
(b)
(c) 7
(d) 8.
(c) 105
(d) N/T
Number of ways in which 5 persons including a sardar can be arranged in a row so that the sardar is always in the middle
25.
10
P5
No. of ways of arranging letters of word “NINETEEN” such that no two E’s are together
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-85 Contact: 9953168795, 9268789880
Content marketed & distributed by FaaDoOEngineers.com (a)
81 (3! )
(c) 26.
2
5! 6 C3 3!
10! 6 !4 !
(c) 7 C 4
29.
30.
31.
32.
33.
(d) N/T
34.
(b) 6 ! (d) N/T
35.
20 persons are to be sitted around a circular table out of these 20 two are brothers. Then number of arrangement in which their will be exactly 3 persons between two brothers is (a) 19C3 x 16! x 5!
28.
(c) 144
5! 8 C3 3!
Number of ways of arranging six ‘+’ & 4 ‘–’ signs can be arranged in a row so that no two ‘–’ signs are together (a)
27.
(b)
(b)
36.
19! 3!2!
(d) 72
A person can select atleast one & atmost n coins from 2n + 1 (distinct) coins in 255 ways, n is equal to (a) 4
(b) 1
(c) 2
(d) 3
Number of 4 digit numbers divisible by 4 can be formed using {1, 2, 3, 4, 5} (a) 24
(b) 30
(c) 125
(d) N/T
No. of ways in which 9 different pearls can be arranged to form a necklace so that 4 particular pearls can’t be separated is (a) ! 5 ! 5
(b) 1440
(c) 9!/4!
(d) N/T
No. of numbers formed by 1,2,3,4,3,2,1 such that odd digits are at odd place is
(c) 18C3 x 3! x 2! x 15! (d) N/T
(a) 430
(b) 36
From two sets of || lines consisting of 6 & 7 lines, no. of possible rectangles using these
(c) 18
(d) N/T
(a) 6 C 2 7 P2
(b)
(c) 6 P2 7 P2
(d) N/T
13
37.
C4
There are 2 points on a line, 3 points on another line and 4 points on yet another line. The total number of triangles that can be formed by joining these points, is (a) 30
(b) 205
(c) 79
(d) 85
38.
39.
Number of words using letters of ALGEBRA so that relative order of vowels & consonants do not change (a) 90
(b) 120
(c) 96
(d) 240
Given line segments of lengths units. Number of s formed is (a) 5C3
(b) 17
(c) 18
(d) 19
40. 2,3,4,5,6,7
Number of six digit numbers formed from digits 1, 2, 3, 4, 5, 6 and 7 so that digits do not repeat and terminal digits are even, is (a) 288
(b) 720
41.
Number of words of letters of ‘BANANA’ in which two N’s do not appear adjacently is : (a) 40
(b) 60
(c) 80
(d) 100
Number of arrangements of letters of word ‘BANANA’ in which all A appear adjacently is (a) 24
(b) 20
(c) 12
(d) 100
The number of arrangements of the letters of the word ‘BANANA’ in which no two A’s appear adjacently is : (a) 48
(b) 60
(c) 12
(d) 36
Ways in which the letters of the word QUADRATIC can be arranged so that no two vowels & no two consonants are together, is (a) 5! 4!
(b) 2X5!4!
(c) 12X5!
(d) 6 C 4 5!4! /2
There are (n + 1) white and (n + 1) black balls each set numbered 1 to n + 1. The number of ways the balls can be arranged in a row so that adjacent balls are of different colours is :
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-85 Contact: 9953168795, 9268789880
Content marketed & distributed by FaaDoOEngineers.com 42.
43.
44.
(a) [ (n + 1) !]2
(b) 2 (2n !)
(c) 2 [ (n + 1) ! ]
(d) 2 [(n + 1) !]2
Sum of all the numbers formed by 1, 3, 5, 7, 9 using all at a time (a) 6666600
(b) 6666000
(c) 5666600
(d) N/T
46.
47.
48.
49.
50.
51.
52.
54.
Sum of all the numbers at hundreds place formed by 1, 3, 5, 7using all at a time (a) 96000
(b) 960
(c) 96
(d) N/T
In a paper there are 10 true – false questions no. of ways in which these can be answered (a) 210
(b) 210 – 1
(c)
(d) N/T
10
P2
In a question paper there are 20 questions each with an alternate, no. of ways in which a student can attempt atleast one question (a) 210 (c)
55.
29
+ 10
(b) 210 - 1 (d) N/T
Number of even divisors of 3600 is
No. of ways in which 200 things can be divided into 50 sets, each of 4 things
(a) 5
(a)
(c) 36 45.
53.
(b) 18 (d) N/T (b) 18
(c) 9
(d) N/T
Sum of all divisors of 4500 is (a) 3 x 4 x 31
(b) 7 x 13 x 156
(c) 7 x 13 x 124
(d) N/T
The number of divisors of 1008 of the form 4n + 2 (n 0, n N), is (a) 5
(b) 6
(c) 11
(d) 7
(c)
(b) 22
(c) 25
(d) N/T
200! 50!
(b)
200! 4 50
(d) N/T
DISTRIBUTION OF IDENTICAL THINGS 56.
57.
Index of highest power of 3 in 50! (a) 16
50
4 50!
Number of odd divisors of 3600 is (a) 45
200!
58.
Ways of setting a 40 marks paper consisting of 12 questions of not less than 2 marks (a) 27C16
(b) 27C13
(c) 27C14
(d) 27C12
No. of ways to give 20 apples to 3 boys, each receiving at least 4 apples is (a)
10
C8
(b) 90
(c)
22
C2
(d) N/T
Number of ways of selecting 12 balls from unlimited Red, white & pink balls
No. of zeros at the end of 200 ! is
(a) 54
(b) 55
(a) 49
(b) 50
(c) 91
(d) N/T
(c) 51
(d) N/T
59.
Sum of all no. greater than 2000 formed by 2,3,4,5 is (a) 93324
(b) 96324
(c) 92324
(d) N/T
No. of non – negative integral solutions of x1 + x2 + x3 = 15 is (a) 134
(b) 135
(c) 136
(d) N/T
Exponent of 4 in 200! Is
If x 0, y 1, z 2 then x + y + z = 15 has ordered soln. (x, y, z) where x,y,z W
(a) 63
(b) 64
(a) 91
(b) 455
(c) 65
(d) 80
(c)
(d) N/T
Total no. of 7 digit numbers the sum of whose digits is even is (a) 9 x 106
(b) 45 x 105
(c) 81 x 105
(d) N/T
60.
61.
17
C15
The number of integral solutions to the equation 2x + y + z = 20 ; x, y, z 0 is : (a) 121
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-85 Contact: 9953168795, 9268789880
(b) 120
Content marketed & distributed by FaaDoOEngineers.com (c) 131 62.
63.
64.
65.
66.
The number of integral solutions of the system of equations t1 + t2 + t3 + t4 + t5 = 20 and t1 + t2 + t3 = 5 where tk 0 is: (a) 211
(b) 336
(c) 331
(d) 241
Number of integral 15< a +b+c 20
(b) 685
(c) 1025
(d) N/T
(a) 9
(b) 44
(c) 16
(d) N/T
of
72.
73.
Number of ways of dividing 15 men & 15 women in 15 couples such that each pair has a man & a woman is (a) 1240
(b) 1225
(c) 30!/ 15!x15!
(d) N/T
74.
A dice is thrown three times then number of ways of getting a sum of 8 is
(a) 50
(b) 84
(c) 126
(d) N/T
75.
76.
Number of ways of giving 16 different things to 3 persons according as A< B < C so that B gets 1 more than A & C 2 more than B is
16! (a) 4!5!7!
(b) 4!5!7!
16! 3!5!8!
(d) N/T.
n
i
j
1 i 1
is equal to
j 1 k 1
(a) 106
(b) 96
(c) 90
(d) N/T
2,6 letter are to be posted in 3 boxes , ways of posting if no box remains empty , is (a) 270
(b) 540
(c) 537
(d) 533
H.C.F of n!, (n+1)!, (n+2)! Is (a) n!
(b) (n+2)!
(c) n
(d) N/T
10 bags have21, 22,…,30 different things resp. No. of ways of taking 10 things from a bag is (a)
31
(c)
31
77.
78.
(b)
C 20
C 20
21
C10
31
C 21
(d) N/T
Number of, ways of painting the faces of a dice numbered 1,2,3,4,5,6 with six different colours, is (c)
(b) 18 (d) N/T
Maximum number of point of intersection of 7 st. lines & 5 circles if 3 lines are parallel & 2 circles are concentric , is
(a) 1
There are 10 stations on a circular track. Number of ways a train can stop at three stations so that no two stations are adjacent
(c)
69.
71.
f : { 1,2,3,4,5} { 1,2,3,4,5} , number of onto functions such that f ( i ) i
(c) 27
68.
solutions
(a) 1245
(a) 45 67.
70.
(d) 130
66
(b) 6! (d) 63
Number of ways in which 9 different objects can be distributed to three different people giving 2, 2 and 5 objects to each of them, is (a) 328
(b) 984
(c) 756
(d) 2268
No. of arrangements of ARTICLE so even places are occupied by consonants is (a) 576
(b) 4 C 3 4!
(c) 2 4!
(d) N/T
Number of ways of forming four letter words using letter of MATHEMATICS (a) 136
(b) 8C4
(c) 2554
(d) N/T
Number of ways in which 8 candidates A1, A2, A3,....,A8 can be ranked such that A3 is always before A4 and A4 always comes before A5, is
(a) n(n+1)(2n+1)/6 (b) n(n+1)/2
(a) 2 x 8P6
(b) 8! /6
(c) n(n+1)( n+2)/6 (d) N/T
(c) 8!
(d) 8! /3
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-85 Contact: 9953168795, 9268789880
Content marketed & distributed by FaaDoOEngineers.com 79.
In 20 books 4 are single volumes and others are 8, 5 and 3 volumes resp. Ways in which all these can be arranged in a shelf so that volumes of same book are not separated is (a)
20 ! 8 !5 !3! 4 !
(c) 20 ! 80.
81.
(b) 27405
(c) 27399
(d) none of these.
In a city no person have identical set of teeth and there in no human person without a tooth. Also no person has more than 32 teeth. If we disregard the shape and size of tooth and consider only the positioning of the teeth, then the maximum population of the city is : 232
(b) 232 – 1 +1
(c) 196
84.
85.
86.
88.
89.
90.
(b) 140 91.
The number of different signals which can be given from 6 flags of different colours taking one or more at a time is : (a) 1958
(b) 1956
(c) 16
(d) 64
92.
If letters of word NUMBER are arranged as in dictionary then rank of the word itself (a) 504
(b) 604
(c) 404
(d) N/T
(a) 33
(b) 35
(c) 38
(d) N/T
Number of s whose vertices are vertices of an octagon but none of whose sides happen to come from sides of the octagon (a) 24
(b) 52
(c) 48
(d) 16
15 boys & 12 girls sit in a row, no. of ways in which between the girls at most 2 boys sit is (b)17! –(
12
C 3 X3!)
(c) 17! – ( 12 C 3 X15!) (d)17!–( 91X2!X15!)
(d) none of these.
(d) 280
Mr. A has x children by his first wife and Mrs.B has (x + 1) children by her first husband. They marry & have children of their own. Now there are total ten children. Assuming that children of same parents don’t fight, maximum no. of fights among children
(a) 17! – 12!X 3!
A student is to answer 10 out of 13 questions such that he must choose at least 4 from first five . Number of choices available to him is : (a) 346
83.
(d) none of these.
(a) 27378
(c) 82.
(b) 7 ! 8 ! 5 ! 3 !
Number of ways of selecting four numbers from 1 to 30 so as to exclude every selection of four consecutive numbers is
(a) 232
87.
(a) 15
(b) 16
(c) 20
(d) 21.
A man has 20 friends and wishes to make as many different parties as he can, then number of his friends he should invite in each party is (a) 9
(b) 11
(c) 8
(d) 10
The number of ways of choosing two integers from 1 to 100 such that their product is multiple of 3 is (a) 110C2 – 33C2
(b) 100C2 – 67C2
(c) 33C2
(d) 67C2
(a) 66
(b) 67
There are two rows ,one behind other, of 5 chairs each . Five couples are to be seated , number of arrangements such that no husband sits in front of or behind his wife
(c) 68
(d) N/T
(a) 120x44
(b)
(c)
(d) 44
Words formed by letters of CIRCLE are arranged as in dictionary then rank of circle is
93.
In a tournament, participants play one game with another, two fell ill, having played 3 games each. If total games played is 84, the number of participants at the beginning was:
If all permutation of word AGAIN are arranged in dictionary then 50th word is (a) NAAGI
(b) NAGAI
(c) NAAGA
(d) NAAIG
94.
10
C 5 x44
10
C 5 x120x44
a + b + c = 100 , a ,b ,c (distinct N ) ,if two of these are odd then number of solutions is (a) 1200
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-85 Contact: 9953168795, 9268789880
(b) 3600
Content marketed & distributed by FaaDoOEngineers.com (c) 1225 95.
(d) N/T
There are 2m horizontal lines equispaced at unit distance & 2n vertical lines equispaced at unit distance. How many rectangles with each side odd can be formed (a) 4 m+ n - 2 (c)
96.
97.
98.
99.
m2
n2
(b) ( m+ n -1) (d) mn (n+ 1) ( m + 1)
n objects are placed in a row , then number of ways of selecting three of these such that no two of them are next to each other (a)
n 3
(c)
n 2
n 3
C2
(b)
C3
(d) N/T
C3
Number of integral solutions of n where n N is (a)
n 3
(c)
n 5
n4
C3
(b)
C5
(d) N/T
x + y + z
C4
In a polygon no three diagonals are concurrent. If total number of points of intersections of diagonals interior to polygon be 70 then number of diagonals is (a) 20
(b) 28
(c) 8
(d) None of these
There are three copies each of 4 different books. The number of ways in which they be arranged on a shelf is: (a)
9! (3! )
(c)
2
12! (3! ) 4
(b)
11! (3! ) 2
(d) None of these
100. A boat is to be manned by eight men of whom 2 can only row on bow side and 3 can only row on stroke side. The number of ways in which the crew can be arranged is (a) 4360
(b) 5930
(c) 5760
(d) None of these
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30
31
32
33
34
35
36
37
38
39
40
d
c
b
a
c
c
c
d
c
a
b
b
a
a
b
c
a
c
c
bc
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
d
a
c
c
c
b
b
b
a
a
d
b
a
d
a
a
a
c
c
a
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
a
b
b
b
a
c
b
a
c
a
b
a
c
b
c
a
d
b
b
a
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
b
c
b
d
b
d
a
d
d
a
d
b
b
d
c
c
a
c
c
c
Nishant Gupta, D-122, Prashant vihar, Rohini, Delhi-85 Contact: 9953168795, 9268789880
