Set 1 – Permutations and Combinations

Homework (A)

1. How many three-digit numbers divisible by 5 can be formed by using any of the digits from 0 to 9 such
that none of the digits can be repeated?

(A) 108 (B) 112

(C) 124 (D) 136

2. How many 5-digit numbers can be formed using the digits 1,2,3, ⋯ ,9 such that no two consecutive digits
are the same?

(A) None of these options (B) 9 × 8

(C) 9 (D) 8

3. In how many different ways can the letters of the word ‘JUDGE’ be arranged such that the vowels
always come together?

(A) None of these options (B) 48

(C) 32 (D) 64

4. In how many ways can the letter of the word ‘LEADER’ be arranged?

(A) None of these options (B) 120

(C) 360 (D) 720

5. How many words can be formed by using all the letters of the word ‘BIHAR’?

(A) 120 (B) 24

(C) 120 (D) 60

6. How many arrangements can be made out of the letters of the word ‘ENGINEERING’?

(A) 924 000 (B) 277 200

(C) None of the options (D) 182 000

7. In how many ways can 5 men draw water from 5 taps if no tap can be used more than once?

(A) None of the options (B) 720

(C) 60 (D) 120

8. In how many ways can 11 software engineers and 10 civil engineers be seated in a row so that they are
positioned alternatively?

(A) 7! × 7! (B) 6! × 7!
(C) 10! × 11! (D) 11! × 11!

9. In how many ways can 10 software engineers and 10 civil engineers be seated in a row so that they are
positioned alternatively?

(A) 2(10!) (B) 2 × 10! × 11!

(C) 10! × 11! (D) (10!)

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10. There are three identical blue marbles and four identical yellow marbles arranged in a row.
(a) How many different arrangements are possible?
(b) How many different arrangements of just five of these marbles are possible?

11. Consider arranging the letters of the word ‘EXPENSIVE’.

(a) How many distinct arrangements are possible if every letter is used?
(b) Find the number of distinct arrangements that can be created using exactly four of the nine letters
available.

12. How many signals can be made using 6 different coloured flags when any number of them can be
hoisted at one time?

(A) 1956 (B) 1720

(C) 2020 (D) 1822

13. There are two books each of 5 volumes and two books each of 2 volumes. In how many ways can these
books be arranged in a shelf so that the volumes of the same book should remain together?

(A) 4! × 5! × 2! (B) 4! × 14!

(C) 14! (D) 4! × 5! × 5! × 2! × 2!

14. In how many ways can 11 people be arranged in a row such that 3 particular people should always be
together?

(A) 9! × 3! (B) 9!
(C) 11! (D) 11! × 3!

15. A company has 11 software engineers and 7 civil engineers. In how many ways can they be seated in a
row so that all the civil engineers are always together?

(A) 18! × 2 (B) 12! × 7!

(C) 11! × 7! (D) 18!

16. How many 8-letter arrangements can be constructed from the English alphabet containing at least 3 Ls?

17. Consider rearranging the letters of the word ‘HULLABALOO’.

(a) How many different ways can the letters be arranged?
(b) How many different ways can the letters be ordered if the three letters, H, U and B are next to each
other in any order?
18. In how many ways can 4 different balls be distributed amongst 5 different boxes when any box can have
any number of balls?

(A) 5 − 1 (B) 5
(C) 4 − 1 (D) 4

19. How many arrangements of the word ‘EQUATION’ are there if none of the consonants are together?

20. A question paper has two parts, and , each containing 10 questions. If a student needs to choose 8
from part and 4 from part , in how many ways can he do this?

(A) None of these (B) 6020

(C) 1200 (D) 9450

21. 5 cards from a standard 52 card deck are chosen and placed face up on a table.
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 (a) In how many different ways can the cards show three-of-a-kind? (i.e. exactly three cards of the same
type, e.g. 3 fives, 3 sevens, 3 kings, etc.)
(b) In how many different ways can the cards show a full house? (i.e. a three-of-a-kind and a pair.)

22. A bag contains 2 white balls, 3 black balls and 4 red balls. In how many ways can 3 balls be drawn from
the bag, if at least one black ball is to be included in the draw?

(A) 64 (B) 128

(C) 32 (D) None of these

23. From a group of 7 men and 6 women, five people are to be selected to form a committee so that at least 3
men are there in the committee. In how many ways can this be done?

(A) 624 (B) 702

(C) 756 (D) 812

24. *The following eight tiles are taken from a scrabble set: , , , , , , , . How many different 4
letter permutations can be formed from these eight tiles?

25. The diagram shows 9 points lying on the plane, 5 of which lie on the line . The remaining 4 points do
not lie on the line and no other set of 3 points are collinear.

(a) How many sets of 3 points can be chosen from the 5 points lying on ?
(b) How many distinct triangles can be formed using any 3 of the 9 points as vertices?

26. *A train is leaving town , heading towards town without turning around. There are 13 train stations
between the two towns.

(a) In how many ways can the train stop at 4 of the 13 stations?
(b) In how many ways can the train stop at 4 of the 13 stations if the train does not stop at
consecutive stations?

27. In how many ways can a team of 5 people be formed out of a total of 10 people such that two particular
people should be included in each team?

(A) 56 (B) 28
(C) 112 (D) 120

28. In how many ways can a team of 5 people be formed out of a total of 10 people such that two particular
people should not be included in any team?

(A) 56 (B) 112

(C) 28 (D) 128

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29. A company has 10 software engineers and 6 civil engineers. In how many ways can a committee of 4
engineers be formed from them such that the committee must contain at least 1 civil engineer?

(A) 1640 (B) 1630

(C) 1620 (D) 1610

30. From a group of 8 women and 6 men, a committee consisting of 3 men and 3 women is to be formed. In
how many ways can the committee be formed if two of the women refuse to serve together?

(A) 1020 (B) 1000

(C) 712 (D) 896

31. From a group of 8 women and 6 men, a committee consisting of 3 men and 3 women is to be formed. In
how many ways can the committee be formed if one man and one woman refuse to serve together?

(A) 722 (B) 910

(C) 612 (D) 896

32. Naresh has 10 friends and he wants to invite 6 of them to a party. How many times will 3 particular
friends always attend the party?

(A) 720 (B) 120

(C) 126 (D) 35

33. Haresh has 10 friends and he wants to invite 6 of them to a party. How many times will 3 particular
friends never attend the party?

(A) 8 (B) 720

(C) 35 (D) 7

34. A photography club consists of 9 female members and 6 male members. A committee of five is to be
selected from the members in the club.
(a) In how many ways can the committee be selected?
(b) In how many ways can the committee be selected if the majority of the committee members are to be
female?
(c) Two particular club members always insist on serving on committees together. If they can’t serve
together, they won’t serve at all. In how many ways can such a committee be selected?

35. An instructor gives an exam with fourteen questions. Students are allowed to choose any ten to answer.
(a) How many different choices of ten questions are there?
(b) Suppose the exam instructors specify that at most one of the first three questions may be included
amongst the ten. How many different choices satisfy this condition?

36. Find the number of triangles that can be formed using 14 points on a plane such that 4 points are
collinear.

(A) 480 (B) 360

(C) 240 (D) 120

37. There are 8 points in a plane out of which 3 are collinear. How many straight lines can be formed by
joining two of these points?

(A) 16 (B) 26
(C) 22 (D) 18
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38. How many quadrilaterals can be formed by joining the vertices of an octagon?

(A) 60 (B) 70
(C) 65 (D) 74

39. How many straight lines can be formed by joining 12 points on a plane out of which no points are
collinear?

(A) 72 (B) 66
(C) 58 (D) 62

40. In a birthday party, every person shakes hands with every other person. If there was a total of 28
handshakes at the party, how many people were present at the party?

(A) 9 (B) 8
(C) 7 (D) 6

41. Find the number of triangles which can be drawn out of given points on a circle.

(A) ( + 1) 1 (B) 1
(C) ( + 1) 3 (D) 3

42. A company has 10 software engineers and 6 civil engineers. In how many ways can a committee of 4
engineers be formed from them such that the committee must contain exactly 1 civil engineer?

(A) 800 (B) 720

(C) 780 (D) 740

Homework (A) Answers

1. D 2. B 3. B 4. C 5. C
6. B 7. D 8. C 9. A
10. .
! ! ! !
(a) ! ! (b) ! !
+ ! !
+ !
= 25
11.
! × ! × !
(a) ! (b) !
+ !
+ 7C4 × 4!
12. A 13. D 14. A 15. B
16. 575 111 451
17.
(a) 151 200 (b) 10 080
18. B
19. 14 400
20. D
21.
(a) 4C3 × 13 × 48C2 (b) 4C3 × 13 × 4C2 × 12
22. A
23. C
24. 606
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25.
(a) 10 (b) 74
26. .
(a) 715 (b) 210
27. A 28. A 29. D 30. B 31. B
32. D 33. D
34.
(a) 15C5 (b) 2142 (c) 13C3 + 13C5
35.
(a) 14C10 (b) 11C9 × 3C1 + 11C10
36. B 37. B 38. B 39. B 40. B
41. D 42. B

Set 2 – Arrangement in a Circle

Homework (B)

1. In how many ways can 10 engineers and 4 doctors be seated at a round table without any restrictions?

(A) 14 10 (B) 14!

(C) 13! (D) None of these options

2. A company has 10 software engineers and 6 civil engineers. In how many ways can they be seated
around a round table so that all the civil engineers are together?

(A) 10! × 6! (B) 11! × 6!

(C) (10!) (D) 9! × 6!

3. In how many ways can 5 girls and 5 boys form a circle such that they boys and the girls alternate?

(A) 2880
(B) 1400
(C) 1200
(D) 3212

4. Kent went to celebrate his 18th birthday by having a dinner party for himself and nine of his friends (five
girls and four boys). In how many ways can the people be seated at a round table if:
(a) The boys and girls are to alternate?
(b) Kent is to be seated between two particular girls?

5. *In how many ways can 11 people sit at a round table for dinner if a particular couple insist on sitting
next to one another and three individuals insist on being separated?

Homework (B) Answers

1. C
2. A
3. A
4.
(a) 4! 5! (b) 2! 7!
5. 302 400
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Set 3 – Two Step Permutation and Combination

Homework (C)

1. For the word ‘DOMESTIC’, how many arrangements of 4 letters can be made if:
(a) The M is included, and the C is not?
(b) The O and M must be included?

2. *Five letters are chosen from the letters of the word ‘WRITING’. These five letters are then placed
alongside one another to form a five-letter arrangement. Find the number of distinct five letter
arrangements which are possible.

Homework (C) Answers

1.
(a) 1 × 6C3 × 4! (b) 1 × 6C2 × 4!
2. 1320

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