Permutation and Combination problems
- 1. How many straight lines can be drawn from 15 non-collinear points?
Options- A. 105
- B. 120
- C. 110
- D. 115 Discuss
Correct Answer: 105
Explanation:
Required number of lines = nC2
= 15C2
= (15 x 14)/2
= 105
- 2. A library has 'a' copies of one book, 'b' copies of each of two book, 'c' copies of each of three books and single copy of 'd' book. The number of ways in which these books can be distribute is?
Options- A. (a + b + c + d)! / (a! b! c!)
- B. (a + 2b + 3c + d)! / {a! (b!)2 (c!)3}
- C. (a + 2b + 3c + d )! / (a! b! c!)
- D. None of these Discuss
Correct Answer: (a + 2b + 3c + d)! / {a! (b!)2 (c!)3}
Explanation:
Total number of books = a + 2b + 3c + d . Since there are 'b' copies of each of two books, 'c' copies of each of three books and single copy of 'd' book.
Therefore, the total number of arrangements is = (a + 2b + 3c + d )! / {a! (b!)2 (c!)3}
- 3. The number of triangles that can be formed by choosing the vertices from a set of 12 points, seven of which lie on the same straight line, is?
Options- A. 185
- B. 175
- C. 115
- D. 105 Discuss
Correct Answer: 185
Explanation:
Required no. of triangles = 12C3 - 7C3 = 185
- 4. How many words of 4 consonants and 3 vowels can be made from 12 consonants and 4 vowels. If all the letters are different?
Options- A. 251820
- B. 258120
- C. 281520
- D. 285120 Discuss
Correct Answer: 285120
Explanation:
Required no. of words = 12P4 x 4P3
= 12 x 11 x 10 x 9 x 4 x 3 x 2
= 120(100 - 1) x 24
= 288000 - 2880
= 285120
- 5. 12 persons are to be arranged to a round table. If two particular persons among them are not to be side by side, the total number of arrangements is?
Options- A. 9 (10!)
- B. 2 (10!)
- C. 45 (8!)
- D. 10! Discuss
Correct Answer: 9 (10!)
Explanation:
12 persons can be seated around a round table in 11! ways. The total number of ways in which 2 particular persons sit side by side = 10! x 2!.
Hence, the required number of arrangements = 11! - 10! x 2! = 9 x (10!)
- 6. The total number of selections of fruit which can be made from 3 bananas, 4 apples and 2 oranges is?
Options- A. 39
- B. 315
- C. 512
- D. None of these Discuss
Correct Answer: None of these
Explanation:
Required number of ways = (2 + 1) (3 + 1) (4 + 1) - 1 = 59
- 7. The total number of arrangements of the letters in the expression a 3b 2c 4 when written at full length is?
Options- A. 1260
- B. 2520
- C. 610
- D. None of these Discuss
Correct Answer: 1260
Explanation:
We have 9 letters 3 a's, 2 b's and 4 c's. These 9 letters can be arranged in 9! / (3! 2! 4!) = 1260 ways.
- 8. If nC r - 1 = 36, nC r = 84 and nC r + 1 = 126 then?
Options- A. n = 8, r = 4
- B. n = 9, r = 3
- C. n = 7, r = 5
- D. None of these Discuss
Correct Answer: n = 9, r = 3
Explanation:
nCr - 1 = 36, nCr = 84 and nCr + 1 = 126
? nCr / nCr - 1 = 84/36
? 3n - 10r + 3 = 0 ....(i)
nCr + 1 / nCr = 126/84
? 2n - 5r - 3 = 0 .............(ii)
From (i) and (ii) r = 3, n = 9
- 9. If 35C n + 7 = 35C 4n - 2, then all the values of n are given by?
Options- A. 28
- B. 3, 6
- C. 3
- D. 6 Discuss
Correct Answer: 3, 6
Explanation:
35Cn + 7 = 35C4n - 2
We know that if nCx = nCy then, Either x = y or x + y = n
Case 1: x + y = n
? n + 7 + 4n - 2 = 35
? 5n + 5 = 35
? 5n = 30
? n = 6
Case 2: x = y
? n + 7 = 4n - 2
? 4n - n = 2 + 7
? 3n = 9
? n = 3
- 10. If 2n + 1P n - 1 : 2n - 1P n = 3 : 5, then n is equal to?
Options- A. 4
- B. 6
- C. 3
- D. 8 Discuss
Correct Answer: 4
Explanation:
We have If 2n + 1Pn - 1 / 2n - 1Pn = 3/5
? 5 x 2n + 1Pn - 1 = 3 x 2n - 1Pn
? {5 x (2n + 1)!} / {(n + 2 )!} = 3 x {(2n - 1)!} / {(n - 1)!}
? 10(2n + 1) = 3(n + 2) (n + 1)
? 3n2 - 11n - 4 = 0
?(3n + 1 ) (n - 4) = 0
? n = 4
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