Permutation and Combination Practice Test Questions – Aptitude -- Free Practice

Question 1

The letters of the word VIOLENT are arranged so that the vowels occupy even place only. The number of permutations is:

A

144✓✗

B

120✓✗

C

24✓✗

D

72✓✗

Explanation

VIOLENT has 7 letters: 3 vowels (I, O, E) and 4 consonants (V, L, N, T).
Of the 7 positions, exactly 3 are even (2nd, 4th, 6th) -- matching the 3 vowels exactly.
Vowels arranged in the 3 even positions: 3! = 6 ways
Consonants arranged in the remaining 4 (odd) positions: 4! = 24 ways
Total = 3! × 4! = 6 × 24 = 144

Question 2

A garden has 6 tall trees in a row. In how many ways can 5 children stand, one in a gap between the trees, in order to pose for a photograph?

A

24✓✗

B

120✓✗

C

720✓✗

D

30✓✗

Explanation

6 trees in a row create 5 gaps between them.
5 children are to be arranged in these 5 gaps: 5! = 120

Question 3

Find 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.

A

720✓✗

B

120✓✗

C

96✓✗

D

None of these✓✗

Explanation

MONDAY has 6 distinct letters.
Arrangements beginning with M: fix M first, arrange remaining 5 letters (O,N,D,A,Y) → 5! = 120
Of these, arrangements that ALSO end with N: fix M first and N last, arrange remaining 4 letters (O,D,A,Y) in between → 4! = 24
Required arrangements = 120 - 24 = 96

Question 4

Five bulbs, of which three are defective, are to be tried in two light-points in a dark room. In how many trials shall the room necessarily be lightened?

A

10✓✗

B

7✓✗

C

3✓✗

D

None of these✓✗

Explanation

Total ways to pick 2 bulbs out of 5 = C(5,2) = 10
Ways where BOTH picked bulbs are defective (room stays dark) = C(3,2) = 3
Ways the room is lightened (at least one working bulb) = 10 - 3 = 7

Question 5

The number of ways of painting the faces of a cube by 6 different colours is:

A

30✓✗

B

36✓✗

C

24✓✗

D

1✓✗

Explanation

With all 6 faces getting distinct colours, the number of ways to assign 6 colours to 6 faces (ignoring orientation) is 6! = 720.
A cube has 24 rotational symmetries, so each distinct painting is counted 24 times.
Distinct paintings = 720 / 24 = 30

Question 6

If from a population with 25 members, a random sample without replacement of 2 members is taken, the number of all such samples is:

A

300✓✗

B

625✓✗

C

50✓✗

D

600✓✗

Explanation

Number of samples = C(25,2) = 25 × 242 = 300

Question 7

A room has 10 doors. In how many ways can a man enter the room by one door and come out by a different door?

A

90✓✗

B

100✓✗

C

50✓✗

D

None of these✓✗

Explanation

He can enter by any of 10 doors, then exit by any of the remaining 9 doors.
Total ways = 10 × 9 = 90

Question 8

There are 12 questions to be answered as Yes or No. In how many ways can this be answered?

A

1021✓✗

B

2048✓✗

C

4096✓✗

D

None of these✓✗

Explanation

Each question has 2 possible answers, independently, over 12 questions:
2¹² = 4096

Question 9

In how many ways can 3 prizes out of 5 be distributed amongst 3 brothers equally?

A

10✓✗

B

45✓✗

C

60✓✗

D

120✓✗

Explanation

First choose which 3 of the 5 prizes are distributed: C(5,3) = 10
Then distribute those 3 prizes among 3 brothers, one each: 3! = 6
Total = 10 × 6 = 60

Question 10

A box contains 7 red, 6 white and 4 blue balls. How many selections of three balls can be made so that none is red?

A

90✓✗

B

120✓✗

C

48✓✗

D

None of these✓✗

Explanation

Non-red balls = 6 white + 4 blue = 10.
Ways to select 3 from these 10 = C(10,3) = 120

Question 11

A user wants to create a password using 4 lowercase letters (a-z) and 3 uppercase letters (A-Z). No letter can be repeated in any form. In how many ways can the password be created if the password must start with an uppercase letter?

A

15 × 26 × 25 × 24 × 23 × 22 × 21 × 20✓✗

B

26 × 25 × 24 × 23 × 22 × 2 × 21✓✗

C

26 × 5 × 25 × 24 × 23 × 2 × 22 × 21✓✗

D

6 × 26 × 25 × 24 × 23 × 22 × 21✓✗

Explanation

No letter may be repeated in either case, so the 7 characters use 7 different letters. Taken in order, these can be chosen in 26 × 25 × 24 × 23 × 22 × 21 × 20 ways.
The first character must be uppercase. The other 2 uppercase characters take 2 of the remaining 6 places: ⁶C₂ = 15 ways.
Total = 15 × 26 × 25 × 24 × 23 × 22 × 21 × 20.

Question 12

A boy has 3 library tickets and 8 books of his interest in the library. Of these 8, he does not want to borrow Mathematics Part-II unless Mathematics Part-I is also borrowed. In how many ways can he choose the three books to be borrowed?

A

41✓✗

B

51✓✗

C

61✓✗

D

71✓✗

Explanation

Total ways to choose 3 books from 8 = C(8,3) = 56
Invalid cases (Part-II chosen but Part-I not): choose Part-II, then the remaining 2 books from the other 6 (excluding both Part-I and Part-II) = C(6,2) = 15
Valid ways = 56 - 15 = 41

Question 13

5 persons are sitting at a round table in such a way that the tallest person is always immediately to the right of the shortest person. The number of such arrangements is

A

6✓✗

B

8✓✗

C

24✓✗

D

None of these✓✗

Explanation

Treat the tallest and shortest persons as one unit (in a fixed relative order) along with the remaining 3 persons: circular arrangements of these 4 units = (4 - 1)! = 6

Question 14

An examination paper with 10 questions consists of 6 questions in Algebra and 4 questions in Geometry. At least one question from each section is to be attempted. In how many ways can this be done?

A

945✓✗

B

100✓✗

C

1000✓✗

D

None of these✓✗

Explanation

Ways to choose at least one question from Algebra = 2⁶ - 1 = 63
Ways to choose at least one question from Geometry = 2⁴ - 1 = 15
Total ways = 63 × 15 = 945

Question 15

If 12 school teams are participating in a quiz contest, then the number of ways the first, second and third positions may be won is

A

1,230✓✗

B

1,320✓✗

C

3,210✓✗

D

None of these✓✗

Explanation

Number of ways = 12 × 11 × 10 = 1,320

Question 16

A polygon has 44 diagonals. The number of sides is:

A

6✓✗

B

7✓✗

C

8✓✗

D

11✓✗

Explanation

Number of diagonals of an n-sided polygon = n(n-3)2
n(n-3)2 = 44
n² - 3n - 88 = 0
n = 3 ± √9 + 3522 = 3 ± 192
n = 11 (taking the positive root)

Question 17

In how many ways can 5 different trophies be arranged on a shelf if one particular trophy must always be in the middle?

A

24✓✗

B

120✓✗

C

48✓✗

D

144✓✗

Explanation

With one trophy fixed in the middle position, the remaining 4 trophies are arranged freely in the remaining 4 positions: 4! = 24

Question 18

If ⁿC₇ = ⁿC₅, the value of 'n' is:

A

10✓✗

B

14✓✗

C

12✓✗

D

None of these✓✗

Explanation

Using ⁿC_r = ⁿC_s ⇒ r + s = n:
7 + 5 = 12

Question 19

How many total combinations can be formed of different counters marked as 1, 2, 3, 4, 5, 6, 7 & 8, taking 4 counters at a time, with at least one odd and one even numbered counter in each combination?

A

68✓✗

B

66✓✗

C

64✓✗

D

62✓✗

Explanation

Total ways to choose 4 counters from 8 = C(8,4) = 70
Subtract combinations that are all-odd (C(4,4) = 1) and all-even (C(4,4) = 1)
Required combinations = 70 - 1 - 1 = 68

Question 20

How many five-digit numbers can be formed with the digits 2, 3, 4, 5, 6, 1 (using any 5 of the 6, without repetition) which are NOT divisible by 5?

A

1200✓✗

B

400✓✗

C

600✓✗

D

1400✓✗

Explanation

Total 5-digit numbers using 5 of the 6 digits = P(6,5) = 720
Numbers divisible by 5 must end in 5: the remaining 4 digits (from the other 5) can be arranged in the first 4 positions in P(5,4) = 120 ways
Numbers NOT divisible by 5 = 720 - 120 = 600

Question 21

How many different groups of 3 people can be formed from a group of 5 people?

A

5✓✗

B

6✓✗

C

10✓✗

D

9✓✗

Explanation

Number of groups = C(5,3) = 10

Question 22

Find the number of ways the letters of the word 'RUBBER' can be arranged?

A

450✓✗

B

362✓✗

C

250✓✗

D

180✓✗

Explanation

The word 'RUBBER' has 6 letters with repetitions: 2 R's and 2 B's. Required number of arrangements = 6!2! × 2! = 7204 = 180.

Question 23

Find in how many different ways, the letters of the word 'LEADING' can be arranged in such way that the vowels always come together?

A

548✓✗

B

426✓✗

C

720✓✗

D

790✓✗

Explanation

'LEADING' has 7 letters: vowels E, A, I and consonants L, D, N, G. Treating the vowels as one block, we have 5 units to arrange: 5! = 120 ways. The 3 vowels within the block can be arranged in 3! = 6 ways. Total ways = 120 × 6 = 720.

Question 24

Find the value of ²⁰C₁₇

A

1260✓✗

B

1140✓✗

C

2580✓✗

D

3200✓✗

Explanation

ⁿC_r = n!r!(n - r)!. ²⁰C₁₇ = 20!17!(20 - 17)! = 20 × 19 × 183 × 2 × 1 = 1140.

Question 25

Out of 5 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?

A

60✓✗

B

1200✓✗

C

5230✓✗

D

7200✓✗

Explanation

3 consonants can be selected from 5 in ⁵C₃ = 10 ways, and 2 vowels from 4 in ⁴C₂ = 6 ways, giving 10 × 6 = 60 groups of 5 letters. Each group of 5 letters can be arranged in 5! = 120 ways. Required number of words = 60 × 120 = 7200.