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- Question
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
- Correct Answer
- 20
ExplanationSince each desired number is divisible by 5, so we must have 5 at the unit place. So, there is 1 way of doing it.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.
Permutation and Combination problems
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- 1. 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:
In the word 'CORPORATION', we treat the vowels OOAIO as one letter.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.
- 2. 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) = ❨ 8 x 7 x 6 x 10 x 9 x 8 x 7 ❩ 3 x 2 x 1 4 x 3 x 2 x 1 = 11760. - 3. 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:
The word 'LEADING' has 7 different letters.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.
- 4. 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:
There are 6 letters in the given word, out of which there are 3 vowels and 3 consonants.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.
- 5. 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:
The word 'LEADER' contains 6 letters, namely 1L, 2E, 1A, 1D and 1R.∴ Required number of ways = 6! = 360. (1!)(2!)(1!)(1!)(1!) - 6. 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 - 7. In a group of 6 boys and 4 girls, four children are to be selected. In how many different ways can they be selected such that at least one boy should be there?
Options- A. 159
- B. 194
- C. 205
- D. 209
- E. None of these Discuss
Correct Answer: 209
Explanation:
We may have (1 boy and 3 girls) or (2 boys and 2 girls) or (3 boys and 1 girl) or (4 boys).∴ Required number
of ways= (6C1 x 4C3) + (6C2 x 4C2) + (6C3 x 4C1) + (6C4) = (6C1 x 4C1) + (6C2 x 4C2) + (6C3 x 4C1) + (6C2) = (6 x 4) + ❨ 6 x 5 x 4 x 3 ❩ + ❨ 6 x 5 x 4 x 4 ❩ + ❨ 6 x 5 ❩ 2 x 1 2 x 1 3 x 2 x 1 2 x 1 = (24 + 90 + 80 + 15) = 209. - 8. In how many different ways can the letters of the word 'MATHEMATICS' be arranged so that the vowels always come together?
Options- A. 10080
- B. 4989600
- C. 120960
- D. None of these Discuss
Correct Answer: 120960
Explanation:
In the word 'MATHEMATICS', we treat the vowels AEAI as one letter.Thus, we have MTHMTCS (AEAI).
Now, we have to arrange 8 letters, out of which M occurs twice, T occurs twice and the rest are different.
∴ Number of ways of arranging these letters = 8! = 10080. (2!)(2!) Now, AEAI has 4 letters in which A occurs 2 times and the rest are different.
Number of ways of arranging these letters = 4! = 12. 2! ∴ Required number of words = (10080 x 12) = 120960.
- 9. How many 4-letter words with or without meaning, can be formed out of the letters of the word, 'LOGARITHMS', if repetition of letters is not allowed?
Options- A. 40
- B. 400
- C. 5040
- D. 2520 Discuss
Correct Answer: 5040
Explanation:
'LOGARITHMS' contains 10 different letters.Required number of words = Number of arrangements of 10 letters, taking 4 at a time. = 10P4 = (10 x 9 x 8 x 7) = 5040. - 10. From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done?
Options- A. 564
- B. 645
- C. 735
- D. 756
- E. None of these Discuss
Correct Answer: 756
Explanation:
We may have (3 men and 2 women) or (4 men and 1 woman) or (5 men only).∴ Required number of ways = (7C3 x 6C2) + (7C4 x 6C1) + (7C5) = ❨ 7 x 6 x 5 x 6 x 5 ❩ + (7C3 x 6C1) + (7C2) 3 x 2 x 1 2 x 1 = 525 + ❨ 7 x 6 x 5 x 6 ❩ + ❨ 7 x 6 ❩ 3 x 2 x 1 2 x 1 = (525 + 210 + 21) = 756.
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