Permutation and Combination€¦ · Permutation and Combination 3 17. The number of six digit...

Permutation and Combination

1

Exercise-1

1. Total number of four digit numbers having all different digits, is equal to

a. 4536 b. 504

c. 5040 d. 720

2. Total number of ‘n’ digit numbers (n>1). Having the property that no two consecutive

digits the same, is equal to

a. 8n b. 9n

c. 19.10n d. None of these

3. Number of zeros at the end of 300! is equal to

a. 75 b. 89

c. 74 d. 98

4. Total number of words that can be formed using the alphabets of the word KUBER, so

that no alphabet is repeated in any of the formed word, is equal to

a. 325 b. 320

c. 240 d. 365

5. Total number of 9 digit numbers that are divisible by 5, is equal to

a. 810 b.

79.10

c. 89.10 d.

718.10

6. Total number of 5 digit numbers having all different digits and divisible by 4 that can

be formed using the digits 1,3,2,6,8,9 , is equal to

a. 192 b. 32

c. 1152 d. 384

7. Total number of 5 digit number. Having all different digits and divisible by 3, that can

be formed using the digits 0,1.2,3,4,5 , is equal to

a.120 b. 213

c. 96 d.216

8. Total number of ways in which a person can put 8 different rings in the figure of his

right hand is equal to

a. 816P b. 8

11P

c. 8

16C d.8

11C


2

9. Total number of divisors of 5880 is equal to

a. 48 b. 24

c. 96 d. 16

10. Total number of divisors of 480, that are of the form 4n+2, 0,n is equal to

a. 2 b. 3

c. 4 d. None of these

11. Total number of words that can be formed using all letters of the word ‘ANSHUMAN’ is

equal to

a.8!

4! b.

8!

5!

c. 8!

2!2! d.

8!

2!

12. Total number of words that can be formed using all letters of the word ‘BRIJESH’ that

neither begins with ‘I’ nor ends with ‘B’ is equal to

a.3720 b. 4920

c. 3600 d. 4800

13. A convex polygon has 44 diagonals. The number of it’s sides is equal to

a.9 b. 10

c.11 d. 12

14. A library has ‘a’ copies of one book, ‘b’ copies each of two books, ‘c’ copies each of

three books, and single copy of ‘d’ books. The total number of ways in which these

books can be arranged in a shelf, is equal to

a.

2 3

2 3 !

! ! !

a b c d

a b c

b.

3

2 3 !

! 2 ! 3 !

a b c d

a b c

c.

3

2 3 !

!

a b c d

c

d.

2 3 !

! 2 ! !

a b c d

a b c

15. Everybody in a room shakes hand with everybody the total number of handshakes is

equal to 153. The total number of persons in the room is equal to

a. 18 b. 19

c. 17 d. 16

16. The number of numbers that are less than 1000 that can be formed using the digits

0,1,2,3,4,5 such that no digit is being repeated in the formed number, is equal to

a. 130 b. 131

c.156 d. 155


3

17. The number of six digit numbers that can be formed, having the property that every

succeeding digit is greater than the preceding digit, is equal to

a.3

9C b. 3

10C

c.3

8P d. 3

10P

18. ‘n’ men and ‘n2’ women are to be seated in a row so that no women sit together. If

n1>n2, then total number of ways in which they can be seated, is equal to

a. 1

1

n

nC b. 1

2 1 2! !n

nC n n

c. 2 11 1 2

! !n

n C n n

d. 1 21

1 2! !nn C

n n

19. There are ‘n’ numbered seats around a round table. Total number of ways in which

n1(n1<n) persons can sit around the round table, is equal to

a. 1

n

nC b. n

n1P

c. 1 1

n

nC d.

1

n

n 1P

20. Total number of ways in which four boys and four girls can be seated around a round

table, so that no two girls sit together, is equal to

a.7! b. 3! 4!

c. 4! 4! d. 3! 3!

21. Three boys of class X, 4 boys of class XI and 5 boys of class XII, sit in a row. Total

number of ways in which these boys can sit so that all the boys of same class sit

together, is equal to

a. 2

3! 4! 5! b. 2

3! 4! 5!

c. 3! 4! 5! d. 2

3! 4! 5!

22. Total number of ways in which the letters of the word ‘MISSISSIPPI’ be arranged, so

that any two ‘Ss are separated, is equal to

a.7350 b. 3675

c.6300 d. None of these

23. The number of ways in which a mixed double game can be arranged amongst nine

married couples so that no husband and his wife play in the same game, is equal to

a.2 2

9 ,7C C b. 2 2 1

9 ,7 ,2C C C

c.2 2

9P ,7P d. 2 2 1

9P ,7P .2P


4

24. A candidate is required to answer 7 out of 10 questions, which are divided into two

groups, each containing 5 questions , He is not permitted to attempt more than 4

questions from each group. Total number of different ways in which the candidate can

answer the paper, is equal to

a.5

3 42, ,5C C b.

5 5

3 42, P . P

c.5 5

3 4.C C d.

5 5

3 4P . P

25. Total number of ways, in which 22 different books can be given to 5 students, so that

two students get 5 books each and all the remaining students get 4 book each, is

equal to

a.

3

22!

3!2!5! 4! b.

2

22!

3! .2!5!

c.22!

3!2!5!4! d. None of these


5

Exercise-2

1. The total number of six digit numbers x1 x2 x3 x4 x5 x6 having the property that x1<x2

x3<x2 x3<x4<x3 x6 is equal to

a.10C6 b.

612C

c. 611C d. None of these

2. The total number of three digit numbers, the sum of whose digits is even, is equal to

a. 450 b. 350

c.250 d. 325

3. If letters of the word ‘KUBER’ are written in all possible orders and arranged as in a

dictionary then rank of the word ‘KUBER’ will be

a. 67

b. 68

c. 65

d. 69

4. In a chess tournament, all participants were to play one game with the other. Two

players fell ill after having played 3 games each. If total number of games played in

the tournament is equal to 84. then total number of participants in the beginning was

equal to

a.10 b.15

c.12 d. 14

5. In a country no two persons have identical set of teeth and there is no person without

a tooth, also no person has more than 32 teeth. If shape and size of tooth is

disregarded and only the position of tooth is considered, then maximum population of

that country can be

a. 322 b.

322 1

c. can't be obtained d. None of these

6. The total number of flags with three horizontal strips, in order, that can be formed

using 2 identical red, 2 identical green and 2 identical white strips, is equal to

a. 4! b. 3. 4!

c. 2. 4! d. None of these

7. The sides AB, BC, CA of a triangle ABC have 3,4,5 interior points respectively on them.

Total number of triangles that can be formed using these points as vertices, is equal to


6

a. 135 b. 145

c. 178 d. 205

8. ‘n’ different toys have to be distributed among ‘n’ children. Total number of ways in

which these toys can be distributed so that exactly one child gets no toy, is equal to

a. n! b. n!nC2

c. (n-1)! nC2 d.n!n-1 C2

9. Total number of non-negative integral solutions of x1+x2+x3=10 is equal to

a. 12C3 b. 10C3

c. 12C2 d. 10C2

10. Total number of positive integral solutions of x1+x2+2x2=15 is equal to

a. 42 b. 70


11. A class contains 3 girls and four boys. Every Saturday five students go on a picnic. a

different group of students is being sent each week. During the picnic, each girl in the

group is given a doll by the accompanying teacher. All possible groups of five have

gone once, the total number of dolls the girls have got is

a. 21 b. 45

c. 27 d. 24

12. Total number of permutations of ‘k’ different things, in a row, taken not more than ‘r’

at a time (each thing may be repeated any number of times) is equal to

a. ' 1k b.

'k

c. ' 1

1

k

k

d.

1

1

rk k

k

13. Total number of 4 digit number that are greater than 3000, that can be formed using

the digits 1,2,3,4,5,6 (no digit is being repeated in any number) is equal to

a.120 b. 240

c.480 d. 80

14. A variable name in certain computer language must be either a alphabet or alphabet

followed by a decimal digit. Total number of different variable names that can exist in

that language is equal to

a. 280 b. 290

c. 286 d. 296

15. Total number of ways of selecting two numbers from the set {1,2,3,4,……….3n} so that

their sum is divisible by 3 is equal to


7

a.

22

2

n n b.

23

2

n n

c.22n n d.

23n n

16. The total number of ways of selecting 10 balls out of an unlimited number of identical

while, red and blue balls is equal to

a.12

2C b.

12

3C

c. 10

2C d. 10

3C

17. Total number of numbers that are less than 3.106 and can be formed using the digits

1,2,3, is equal to

a. 9 813 4.3

2 b. 91

3 32

c. 817.3 3

2 d. 9 81

3 3 32

18. There are 10 person among whom two are brothers. The total number of ways in which

these persons can be seated around a round table so that exactly one person sit

between the brothers, is equal to

a. 2! 7! b. 2! 8!

c. 3! 7! d. 3! 8!

19. A teacher takes 3 children from her class to the zoo at a time as often as she can, but

she doesn’t take the same set of three children more than once. She finds out that she

goes to the zoo 84 times more than a particular child goes to the zoo, Total number of

students in her class in equal to

a. 12 b. 14

c. 10 d. 11

20. A person predicts the outcome of 20 cricket matches of his home team. Each match

can result either in a win, loss or tie for the home team. Total number of ways in

which he can make the predictions so that exactly 10 predictions are correct, is equal

to

a. 20 10

10.2C b. 20 20

10.3C

c. 20 10

10.3C d. 20 30

10.2C

21. Total number of ways in which 15 identical blankets can be distributed among 4 persons

so that each of them gets atleast two blankets, equal to

a. 10

3C b.

9

3C


8

c. 11

3C d. None of these

22. A team of four students is to be selected from a total of 12 students. Total number of

ways in which team can be selected such that two particular students refuse to be

together and other two particular students wish to be together only, is equal to

a. 220 b.182 c.226 d. None of these

23. Two players P1 and P2 play a series of ‘2n’ games. Each game can result in either a win

or loss for P1, Total number of ways in which P1 can win the series of these games, is

equal to

a. 2 212

2

n n

nC

b. 2 212 2.

2

n n

nC

c. 212

2

n n

nC d. 21

2 2.2

n n

nC

24. The total number of 3 letters words that can be formed from the letters of the word

‘SAHARANPUR’. is equal to

a.210 b. 237

c. 247 d. 227

25. 15 identical balls have to be put in 5 different boxes. Each box can contain any number

of balls. Total number of ways of putting the balls in to box so that each box contains

atleast 2 balls, is equal to

a.9

5C b.

10

5C

c. 6

5C d.

10

6C


9

Exercise-3

1. Total number of divisors of n=25, 34,510, 76 that are of the form 4 2, 1 is equal to

a. 385 b.55 c. 384 d. 54

2. Total number of divisors of n=35, 57,79 that are of the form 4 1, 0 , is equal to

a. 240 b.30 c. 120 d. 15

3. Total number of positive integral solutions of the equation x1, x2, x3=60, is equal to

a.27 b. 54 c. 64 d. None of these

4. n1 and n2 are five digits numbers. Total numbers of ways of forming n1 and n2 so that

these numbers can be added without carrying at any stage, is equal to

a. 4

36. 55 b. 4

45 55 c. 5

55 d. None of these

5. n1 and n2 are five digits numbers. Total numbers of ways of forming n1 and n2, so that

these numbers can be added without carrying at any stage, is equal to

a. 3

36 55 b. 3

45 55 c. 4

55 d. None of these

6. Total number of positive integral solutions of x1+x2+x3 10, is equal to

a. 311C b. 39C c.12

3C d.10

3C

7. Total number of positive integral solutions of 15<x1+x2+x3 20, is equal to

a.1125 b. 1150 c. 1245 d. 685

8. ‘n’ is selected from the set {1,2,3,………………100} and the number 26+3n+5n is formed

and the number 2n+3n+5n is formed Total number of ways of selecting ‘n’ so that the

formed number is divisible by 4, is equal to

a. 50 b. 49 c. 48 d. None of these

9. Total number of times, the digit ‘3’ will be written when the integers having less than 4

digits are listed, is equal to

a.300 b. 310 c.302 d. 306

10. : 1,2,3,4,5 , ,f x y t Total number of on the functions ‘f’ is equal to

a.242 b. 245 c.1024 d. 240


10

11. Let f: {1,2,3,4,5}{1,2,3,4,5} Total number of functions ‘f’ that are on to and f (i) i,

is equal to

a. 9 b.44 c. 16 d. None of these

12. Total number of parallelogram that can be formed using ‘n’ parallel lines in one

direction and ‘n2’ parallel lines in another direction is equal to

a. 1 2

2 2

n nC C b. 1 2

2

n n C c. 1 2

1

n n C d. None of these

13. Total number of ways of selecting 2 white sequence on a normal chess board, so that

they don’t belong to the same row or column, is equal to

a. 96 b. 400 c. 480 d. 491

14. Total number of ways of selecting 3 small squares on a normal chess board so that

they don’t belong to the same row, column or diagonal line, is square to

a. 1824 b. 920 c. 392 d. None of these

15. 10 different toys are to be distributed among 10 children. Total number of ways of

distributing these toys so that exactly 2 children do not get any toy, is equal to

a.

2

5

1 110!

3!2!7! 2! 6!

b.

2

5

1 110!

3!2!7! 2! 6!

c.

2

5

1 110!

3!7! 2! 6!

d.

2

4

1 110!

3!7! 2! 6!

16. Total number of regions in which ‘n’ coplanar lines can divide the plane, it is known

that no two lines are parallel and no three of them are concurrent, is equal to

a. 212

2n n b. 21

32

n n

c. 213

2n n d. 2 2n n

17. Total number of regions in which ‘n’ coplanar circles can divide the plane, it is known

that each pair of circles intersect in two different points and no three of them have

common point of intersection, is equal to

a. 212

2n n b. 21

32

n n


11

c. 213

2n n d. 2 2n n

18. Let x1, x2……………………………….xk are the divisors of positive integer ‘n’ (including 1 and

n) If x1+x2+……………………………+xk=75, then 1

1

i ix

will be equal to

a.

2

75

k b.

75

k

c. 75

n d.

2

75

n

19. Let A={x1,x2,x3…………………………., x7} B=={y1,y2,y3}. The total number of functions

f: AB that are onto and there are exactly three element x in A such that f (x)=y2, is

equal to

a. 7

214. C b.

7

314. C

c. 7

27. C d.

7

27. C

20. Total number of polynomials of the form x3+ax2+bx+c that the divisible by x2+1,

where a,b,c {1,2,3,………….9,10} is equal to

a. 10 b. 15

c. 5 d. 8

21. Total number of integers ‘n’ such that 2 n 2000 and H.C.F of n and 36 is one, is

equal to

a. 666 b. 667


22. The number of ways in which three distinct numbers in A.P. can be selected from the

set 1,2,3,...........24 , is equal to

a. 66 b. 132


23. Among the 8! Permutations of the digits 1,2,3,………….8, consider those arrangements

which have the following property. If we take any five consecutive positions, the

product any five consecutive positions, the product of the corresponding digits is

divisible by 5. The number of such arrangement will be equal to

a. 7 b. 2, (7!)

c. 7

4C d. None of these


12

24. The total number of functions ‘f’ from the set {1,2,3} into the set {1,2,3,4,5} such

that f(i) , ,f j j is equal to

a. 35 b. 30

c. 50 d. 60

25. Ten persons numbered 1,2, …………10 play a chess tournament, each player playing

against every other player exactly one game. It is known that no game ends in a draw.

Let w1, w2,………………w10 be the number of games won by players 1,2,3…………….10

respectively and 1 2 10, ,............... be the number of games lost by the players 1,2,

……….10 respectively. Then

a.2 2

1 181w b.

2 2

1 181w

c. 2 2

1 1w d. None of these


13

Answers Exercise-1 Binomial Theorem

1. a 2. b 3. c 4. a 5. d 6. a 7. d 8. b 9. a 10. c

11. c 12. a 13. c 14.a 15. a 16.b 17. a 18. d 19. b 20. b

21. a 22. a 23. b 24. b 25. a

Answers Exercise-2 Complex Number

1. c 2. a 3. a 4. b 5. b 6. a 7. d 8. b 9. c 10. a

11. b 12. d 13. b 14.c 15. b 16.a 17. c 18. b 19. c 20. a

21. a 22. c 23. b 24. c 25. a

Answers Exercise-3 Complex Number

1. c 2. a 3. b 4. a 5. b 6. d 7. d 8. b 9. a 10. d

11. b 12. a 13. b 14.a 15. b 16. a 17. d 18. c 19. b 20. a

21. a 22. b 23. b 24. a 25. c


14


15


16


17

Answers Exercise-1 Quadratic Equation and Expressions

1. a 2. a 3. b 4. a 5. b 6. d 7. d 8. b 9. a 10. b

11. b 12. c 13. b 14.d 15. d 16.b 17. d 18. b 19. d 20. a

21. c 22. a 23. c 24. b 25. c


1. d 2. a 3. d 4. b 5. c 6. b 7. a 8. a 9. b 10. b

11. d 12. c 13. c 14.b 15. c 16.b 17. b 18. d 19. b 20. a

21. a 22. a 23. d 24. c 25. c



18

1. d 2. b 3. a 4. b 5. c 6. a 7. c 8. d 9. a 10. b

11. c 12. a 13. a 14.a 15. c 16. a 17. b 18. c 19. a 20. d

21. d 22. a 23. b 24. b 25. a

Permutation and Combination€¦ · Permutation and Combination 3 17. The number of six digit...

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