Permutation and Combination problems
- 1. If there are 12 persons in a party and each of them shakes hands with each other, how many handshakes do happen in the party?
Options- A. 77
- B. 66
- C. 44
- D. 55 Discuss
Correct Answer: 66
Explanation:
first person will shake hands with 11 other persons. Seconds person will shake hands with 10 other persons, Third person will shake hands with 9 other person and so on.
So , total handshake
= 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = 66
OR
2 person are to be chosen from 12 person to have a hand shake
? Possible ways = 12C2 = 12! / 2!(12 - 2)! = 66
- 2. Find the number of ways of preparing a chain with 5 different coloured beads?
Options- A. 18
- B. 24
- C. 12
- D. 30 Discuss
Correct Answer: 12
Explanation:
Neglecting the direction of beads in the chain, number of ways of preparing a chain with 5 different coloured beads
= (1/2) x (5 -1)! = (1/2) x 4! = 24/2 = 12
- 3. Find the number of different ways of forming a committee consisting of 3 men and 3 women from 6 men and 5 women.?
Options- A. 30
- B. 20
- C. 10
- D. 25 Discuss
Correct Answer: 30
Explanation:
The number of ways of selecting 3 men and 3 women out of 6 men and 5 women = 6C3 x 5C3
= 6!/(3! x 3!) + 5/(3! x 2!)
= 20 + 10 = 30
- 4. Two series of a question booklet for an aptitude test are to be given to twelve students. In how many ways can the students be placed in two rows of six each, so that there should be no identical series side by side and that the students sitting one behind the other should have the same series?
Options- A. 2 x 12C6 x (6!)2
- B. 6! x 6!
- C. 7! x 7!
- D. None of these Discuss
Correct Answer: 6! x 6!
Explanation:
As, these are two sets of booklets, so number of booklet in each set is 6 and this can be arrange in 6! ways.
Also, the other 6 booklets or 2nd set can also be arranged in other 6 students in 6! ways.
? Required number of ways = 6! x 6!
- 5. A department had 8 male and female employees each. A project team involving 3 male and 3 female members needs to be chosen from the department employees. How many different projects teams can be chosen?
Options- A. 112896
- B. 3136
- C. 720
- D. 112 Discuss
Correct Answer: 3136
Explanation:
Total ways = 8C3 x 8C3
= (8 x 7 x 6)/(3 x 2) x (8 x 7 x 6)/(3 x 2)
= 56 x 56 = 3136
- 6. A man has 9 friends, 4 boys and 5 girls. In how many ways can he invite them, if there have to be exactly 3 girls in the invitees?
Options- A. 320
- B. 160
- C. 80
- D. 200 Discuss
Correct Answer: 160
Explanation:
3 girls can be be selected out of 5 girls in 5C3 ways. Since, number of boys to be invited is not given, hence out of 4 boys, he can invite them in (2)4 ways.
? Required number of ways 5C3 x (2)4 = 10 x 16 = 160
- 7. In how many different ways can four books A, B, C and D be arranged one above another in a vertical order such that the books A and B are never in continuous position?
Options- A. 9
- B. 12
- C. 14
- D. 18 Discuss
Correct Answer: 12
Explanation:
The number of arrangement in which A and B are not together
= Total number of arrangement - Number of arrangement in which A and B are together
= 4! - 3! x 2! = 24 - 12 = 12
- 8. Boxes numbered 1, 2, 3, 4 and 5 are kept in a row and they are to be filled with either a red or a blue ball such that no two adjacent boxes can be filled with blue balls Then, how different arrangements are possible, given that all balls of a given colour are exactly identical in all respects?
Options- A. 8
- B. 10
- C. 15
- D. 22 Discuss
Correct Answer: 22
Explanation:
Total number of ways filling the 5 boxes numbered as (1, 2, 3, 4 and 5) with either blue or red balls = 25 = 32
Two adjacent boxes with blue balls can be obtained in 4 ways, i.e., (12), (23), (34) and (45). Three adjacent boxes with blue balls can be obtained in 3 ways i.e., (123), (234) and (345). Four adjacent boxes with blue balls can be obtained in 2 ways i.e., (1234) and (2345) and five boxes with blue balls can be got in 1 way.
Hence, the total number of ways of filling the boxes such that adjacent boxes have blue balls
= (4 + 3 + 2 + 1)
= 10
Hence, the number of ways of filling up the boxes such that no two adjacent boxes have blue balls
= 32 - 10
= 22
- 9. A child has four pockets and three marbles. In how many ways, the child can put the marbles in the pockets?
Options- A. 12
- B. 64
- C. 256
- D. 60 Discuss
Correct Answer: 64
Explanation:
The first marble can be put into the pockets in 4 ways, so can the second and third.
Thus, the number of ways in which the child can put the marbles = 4 x 4 x 4 = 64 ways .
- 10. In how many ways, can the letters of the word 'ASSASSINATION' be arranged, so that all the S are together?
Options- A. 10!
- B. 14! (4!)
- C. 151200
- D. 3628800 Discuss
Correct Answer: 151200
Explanation:
When All S are taken together, then ASSASSINATION has 10 letters .
\
So, 10 letters in total can be arranged in 10 ! ways .
[? All 'S' are considered as 1]
But, here are 3 'A' and 2'I' and 2 'N' .
? Required number of ways = 10 ! / (3 ! x 2 ! x 2 !) = 151200
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