Permutation and Combination problems
- 1. In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together?
Options- A. 120
- B. 720
- C. 4320
- D. 2160
- E. None of these Discuss
Correct Answer: 720
Explanation:
When the vowels OIA are always together, they can be supposed to form one letter.
Then, we have to arrange the letters PTCL (OIA).
Now, 5 letters can be arranged in 5! = 120 ways.
The vowels (OIA) can be arranged among themselves in 3! = 6 ways.
∴ Required number of ways = (120 x 6) = 720.
- 2. A box contains 2 white balls, 3 black balls and 4 red balls. In how many ways can 3 balls be drawn from the box, if at least one black ball is to be included in the draw?
Options- A. 32
- B. 48
- C. 64
- D. 96
- E. None of these Discuss
Correct Answer: 64
Explanation:
| ∴ Required number of ways | = (3C1 x 6C2) + (3C2 x 6C1) + (3C3) | |||||||||||||
|
||||||||||||||
| = (45 + 18 + 1) | ||||||||||||||
| = 64. |
- 3. Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?
Options- A. 210
- B. 1050
- C. 25200
- D. 21400
- E. None of these Discuss
Correct Answer: 25200
Explanation:
| = (7C3 x 4C2) | |||||||||
|
|||||||||
| = 210. |
Number of groups, each having 3 consonants and 2 vowels = 210.
Each group contains 5 letters.
| Number of ways of arranging 5 letters among themselves |
= 5! |
| = 5 x 4 x 3 x 2 x 1 | |
| = 120. |
∴ Required number of ways = (210 x 120) = 25200.
- 4. In how many ways can the letters of the word 'LEADER' be arranged?
Options- A. 72
- B. 144
- C. 360
- D. 720
- E. None of these Discuss
Correct Answer: 360
Explanation:
| ∴ Required number of ways = | 6! | = 360. |
| (1!)(2!)(1!)(1!)(1!) |
- 5. In how many different ways can the letters of the word 'DETAIL' be arranged in such a way that the vowels occupy only the odd positions?
Options- A. 32
- B. 48
- C. 36
- D. 60
- E. 120 Discuss
Correct Answer: 36
Explanation:
Let us mark these positions as under:
(1) (2) (3) (4) (5) (6)
Now, 3 vowels can be placed at any of the three places out 4, marked 1, 3, 5.
Number of ways of arranging the vowels = 3P3 = 3! = 6.
Also, the 3 consonants can be arranged at the remaining 3 positions.
Number of ways of these arrangements = 3P3 = 3! = 6.
Total number of ways = (6 x 6) = 36.
- 6. In how many different ways can the letters of the word 'LEADING' be arranged in such a way that the vowels always come together?
Options- A. 360
- B. 480
- C. 720
- D. 5040
- E. None of these Discuss
Correct Answer: 720
Explanation:
When the vowels EAI are always together, they can be supposed to form one letter.
Then, we have to arrange the letters LNDG (EAI).
Now, 5 (4 + 1 = 5) letters can be arranged in 5! = 120 ways.
The vowels (EAI) can be arranged among themselves in 3! = 6 ways.
∴ Required number of ways = (120 x 6) = 720.
- 7. In how many ways a committee, consisting of 5 men and 6 women can be formed from 8 men and 10 women?
Options- A. 266
- B. 5040
- C. 11760
- D. 86400
- E. None of these Discuss
Correct Answer: 11760
Explanation:
| Required number of ways | = (8C5 x 10C6) | |||||||
| = (8C3 x 10C4) | ||||||||
|
||||||||
| = 11760. |
- 8. In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together?
Options- A. 810
- B. 1440
- C. 2880
- D. 50400
- E. 5760 Discuss
Correct Answer: 50400
Explanation:
Thus, we have CRPRTN (OOAIO).
This has 7 (6 + 1) letters of which R occurs 2 times and the rest are different.
| Number of ways arranging these letters = | 7! | = 2520. |
| 2! |
Now, 5 vowels in which O occurs 3 times and the rest are different, can be arranged
| in | 5! | = 20 ways. |
| 3! |
∴ Required number of ways = (2520 x 20) = 50400.
- 9. How many 3-digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9, which are divisible by 5 and none of the digits is repeated?
Options- A. 5
- B. 10
- C. 15
- D. 20 Discuss
Correct Answer: 20
Explanation:
The tens place can now be filled by any of the remaining 5 digits (2, 3, 6, 7, 9). So, there are 5 ways of filling the tens place.
The hundreds place can now be filled by any of the remaining 4 digits. So, there are 4 ways of filling it.
∴ Required number of numbers = (1 x 5 x 4) = 20.
- 10. In how many ways can a group of 5 men and 2 women be made out of a total of 7 men and 3 women?
Options- A. 63
- B. 90
- C. 126
- D. 45
- E. 135 Discuss
Correct Answer: 63
Explanation:
| Required number of ways = (7C5 x 3C2) = (7C2 x 3C1) = | ❨ | 7 x 6 | x 3 | ❩ | = 63. |
| 2 x 1 |
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