Permutation and Combination problems
- 1. 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:
| ∴ Required number of ways |
= (6C1 x 4C3) + (6C2 x 4C2) + (6C3 x 4C1) + (6C4) | |||||||||||||||||||
| = (6C1 x 4C1) + (6C2 x 4C2) + (6C3 x 4C1) + (6C2) | ||||||||||||||||||||
|
||||||||||||||||||||
| = (24 + 90 + 80 + 15) | ||||||||||||||||||||
| = 209. |
- 2. 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:
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.
- 3. 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:
| Required number of words | = Number of arrangements of 10 letters, taking 4 at a time. |
| = 10P4 | |
| = (10 x 9 x 8 x 7) | |
| = 5040. |
- 4. 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:
| ∴ Required number of ways | = (7C3 x 6C2) + (7C4 x 6C1) + (7C5) | |||||||||||
|
||||||||||||
|
||||||||||||
| = (525 + 210 + 21) | ||||||||||||
| = 756. |
- 5. If 56P r + 6 : 54P r + 3 = 30800 : 1 then the value of r is?
Options- A. 40
- B. 41
- C. 42
- D. None of these Discuss
Correct Answer: 41
Explanation:
? 56Pr + 6 : 54Pr + 3 = 30800 : 1
? 56! / (50 - r)! = (30800 x 54!) / (51-r!)
? 56 x 55 = 30800/(51 - r)
? 51 - r = 10
? r = 41
- 6. How many groups of 6 persons can be formed from 8 men and 7 women?
Options- A. 5000
- B. 5005
- C. 5050
- D. None of these Discuss
Correct Answer: 5005
Explanation:
Total no . of person = 8 + 7 = 15
No. of groups = 15C6 = 15! / {6! (15 - 6)!} = 15! / (6! 9!)
= (15 x 14 x 13 x 12 x 11 x 10) / (6 x 5 x 4 x 3 x 2 x 1)
= 5005
- 7. There are 10 oranges in a basket. Find the no. of ways in which 3 oranges are chosen from the basket?
Options- A. 125
- B. 140
- C. 110
- D. 120 Discuss
Correct Answer: 120
Explanation:
Required number of ways = 10C3
= 10!/ {3! (10 - 3)!} = 10! / (3! 7!) = (10 x 9 x 8) / (3 x 2) = 120
- 8. There are 25 students in a class. Find the number of ways in which a committee of 3 students is to be formed.?
Options- A. 2200
- B. 2300
- C. 2400
- D. 3200 Discuss
Correct Answer: 2300
Explanation:
Required number of ways = 25C3
= (25 x 24 x 23) / (1 x 2 x 3) = 2300
- 9. 8 men entered a lounge simultaneously. If each person shook hands with the other, then find the total no. of hand shakes?
Options- A. 16
- B. 36
- C. 56
- D. 28 Discuss
Correct Answer: 28
Explanation:
Required no. of hand shakes = {8 x (8 - 1)}/2 = 28
- 10. If n+2C 8 : n- 2P 4 = 57 : 16 then the value of n is?
Options- A. 20
- B. 19
- C. 18
- D. 17 Discuss
Correct Answer: 19
Explanation:
n+2C8 : n-2P4 = 57 : 16
? {(n + 2 )! (n - 6)!} / {(n - 6)! (n - 2)! 8!} = 57/16
? (n + 2) (n + 1) n(n - 1) = 143640
? (n2 + n - 2) (n2 + n ) = 143640
? (n2 + n )2 - 2(n2 + n) + 1 = 143641
? (n2 + n - 1)2 = (379)2
[? n2 + n - 1 > 0]
? n2 + n - 1 = 379
? n2 + n - 380 = 0
? (n + 20) (n - 19) = 0
?n = 19 ( Since n is not negative)
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