# Permutations and Combinations Questions and Answers – Set 2

This set of Aptitude Questions and Answers (MCQs) focuses on “Permutations and Combinations – Set 2”.

1. In how many ways a team consisting of 4 women and 3 men can be formed from 8 women and 6 men?
a) 1400
b) 720
c) 1080
d) 1440

Explanation: Given, there are 8 women and 6 men.
4 out of 8 women and 3 out of 6 women.
Number of ways required = (nCr * nCr). Where nCr = n! / r!(n – r)!.
(8C4 * 6C3) = (8! / 4!(8 – 4)! * 6! / 3!(6 – 3)!) = ((8 * 7 * 6 * 5 * 4!) / (4! * 4!) * (6 * 5 * 4 * 3!) / (3! * 3!))
(70 * 20) = 1400
Number of ways the team can be arranged is = 1400.

2. Out of 6 consonants and 5 vowels, how many words can be formed using 4 consonants and 2 vowels?
a) 10800
b) 252000
c) 25200
d) 14400

Explanation: Given that, 4 out of 6 consonants and 2 out of 5 vowels.
So, (nCr * nCr). Where nCr = n! / r!(n – r)!.
(7C4 * 5C2) = (7! / 4!(7 – 4)! * 5! / 2!(5 – 2)!) = (7 * 6 * 5 * 4!) / (4! * 3!) * (5 * 4 * 3!) / (2! * 3!)).
(35 * 10) = 350.
Number of words, each having 4 consonants and 2 vowels = 350.
Each word contains 6 letters.
Number of ways of arranging 6 letters among themselves = 6! = 720.
Required number of ways = (350 * 720) = 252000.

3. In how many ways a group of 4 men and 4 women can be made out of 8 men and 6 women?
a) 1250
b) 720
c) 1050
d) 360

Explanation: Given, 4 out of 8 men and 4 out of 6 women.
So, (nCr * nCr). Where nCr = n! / r!(n – r)!.
(8C4 * 6C4) = (8! / 4! (8 – 4)! * 6! / 4! (6 – 4)!) = (8 * 7 * 6 * 5 * 4!) / (4! * 4!) * (6 * 5 * 4!) / (4! * 2!)).
Number of ways required = 70 * 15 = 1050.

4. In how many ways a group consisting of 4 men and 3 women can be formed from 6 men and 5 women?
a) 60
b) 120
c) 240
d) 150

Explanation: Given, there are 6 men and 5 women.
4 out of 6 men and 3 out of 5 women.
Number of ways required = (nCr * nCr). Where nCr = n! / r!(n – r)!.
(6C4 * 5C3) = (6! / 4!(6 – 4)! * 5! / 3!(5 – 3)!) = ((6 * 5 * 4!) / (4! * 2!) * (5 * 4 * 3!) / (3! * 2!)).
(15 * 10) = 150.
Number of ways the team can be arranged is = 150.

5. From 5 vowels and 4 consonants, how many words can be formed using 3 vowels and 2 consonants?
a) 7200
b) 2400
c) 3600
d) 5600

Explanation: Given that, 3 out of 5 vowels and 2 out of 4 consonants.
So, (nCr * nCr). Where nCr = n! / r!(n – r)!.
(5C3 * 4C2) = (5! / 3!(5 – 3)! * 4! / 2! (4 – 2)!) = (5 * 4 * 3!) / (3! * 2!) * (4 * 3 * 2!) / (2! * 2!)).
(10 * 6) = 60.
Number of words, each having 3 vowels and 2 consonants = 60.
Each word contains 5 letters.
Number of ways of arranging 6 letters among themselves = 5! = 120.
Required number of ways = (60 * 120) = 7200.

6. In how many ways we can select 5 members from a group of 9 people?
a) 120
b) 720
c) 360
d) 240

Explanation: Given that, we have to select 5 members among 9 people.
So, nCr = n! / r!(n – r)!.
9C5 = 9! / 5!(9 – 5)! = 9 * 8 * 7 * 6 * 5! / (5! * 4!) = 126.

7. In how many ways we can select 8 people out of 10 people?
a) 10C6
b) 8C2
c) 8C10
d) 10C8

Explanation: we have to select 8 people out of 10.
nCr is the formula.
Number of ways to select 8 people out of 10 = 10C8.

8. If nPr = 5040 and nCr = 42 then find n and r.
a) 6, 5
b) 7, 5
c) 8, 4
d) 7, 4

Explanation: Given, nPr = 5040 and nCr = 42.
nPr = n! / (n – r)!.
nCr = n! / r!(n – r)!.
In order to find n and r we have to divide nPr with nCr.
nPr / nCr = 5040 / 42 = 120.
r! = 120.
Hence r = 5, now n! / (n – 5)! = 5040.
n(n – 1)(n – 2)(n – 3)(n – 4)(n – 5)(n – 6) = 7 * 6 * 5 * 4 * 3 * 2 * 1

9. The formula for combination is nCr = n! / r!(n – r)!.
a) True
b) False

Explanation: nCr = n! / r!(n – r)!.
Here, ‘n’ is the number of items.
‘r’ = how many items are taken at a time.

10. In how many ways a group of 1 man and 2 women can be made out of 4 men and 8 women?
a) 112
b) 96
c) 84
d) 124

Explanation: Given, there are 6 men and 5 women.
1 out of 4 men and 2 out of 8 women.
Number of ways required = (nCr * nCr). Where nCr = n! / r!(n – r)!.
(4C1 * 8C2) = (4! / 1! (4 – 1)! * 8! / 2! (8 – 2)!) = ((4 * 3!) / (1! * 3!) * (8 * 7 * 6!) / (2! * 6!)).
(4 * 28) = 112.
Number of ways the team can be arranged is = 112.

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