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

View Answer

Explanation: Given, there are 8 women and 6 men.

4 out of 8 women and 3 out of 6 women.

Number of ways required = (

^{n}C

_{r}*

^{n}C

_{r}). Where

^{n}C

_{r}= n! / r!(n – r)!.

(

^{8}C

_{4}*

^{6}C

_{3}) = (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

View Answer

Explanation: Given that, 4 out of 6 consonants and 2 out of 5 vowels.

So, (

^{n}C

_{r}*

^{n}C

_{r}). Where

^{n}C

_{r}= n! / r!(n – r)!.

(

^{7}C

_{4}*

^{5}C

_{2}) = (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

View Answer

Explanation: Given, 4 out of 8 men and 4 out of 6 women.

So, (

^{n}C

_{r}*

^{n}C

_{r}). Where

^{n}C

_{r}= n! / r!(n – r)!.

(

^{8}C

_{4}*

^{6}C

_{4}) = (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

View Answer

Explanation: Given, there are 6 men and 5 women.

4 out of 6 men and 3 out of 5 women.

Number of ways required = (

^{n}C

_{r}*

^{n}C

_{r}). Where

^{n}C

_{r}= n! / r!(n – r)!.

(

^{6}C

_{4}*

^{5}C

_{3}) = (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

View Answer

Explanation: Given that, 3 out of 5 vowels and 2 out of 4 consonants.

So, (

^{n}C

_{r}*

^{n}C

_{r}). Where

^{n}C

_{r}= n! / r!(n – r)!.

(

^{5}C

_{3}*

^{4}C

_{2}) = (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

View Answer

Explanation: Given that, we have to select 5 members among 9 people.

So,

^{n}C

_{r}= n! / r!(n – r)!.

^{9}C

_{5}= 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) ^{10}C_{6}

b) ^{8}C_{2}

c) ^{8}C_{10}

d) ^{10}C_{8}

View Answer

Explanation: we have to select 8 people out of 10.

^{n}C

_{r}is the formula.

Number of ways to select 8 people out of 10 =

^{10}C

_{8}.

8. If ^{n}P_{r} = 5040 and ^{n}C_{r} = 42 then find n and r.

a) 6, 5

b) 7, 5

c) 8, 4

d) 7, 4

View Answer

Explanation: Given,

^{n}P

_{r}= 5040 and

^{n}C

_{r}= 42.

^{n}P

_{r}= n! / (n – r)!.

^{n}C

_{r}= n! / r!(n – r)!.

In order to find n and r we have to divide

^{n}P

_{r}with

^{n}C

_{r}.

^{n}P

_{r}/

^{n}C

_{r}= 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 ^{n}C_{r} = n! / r!(n – r)!.

a) True

b) False

View Answer

Explanation:

^{n}C

_{r}= 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

View Answer

Explanation: Given, there are 6 men and 5 women.

1 out of 4 men and 2 out of 8 women.

Number of ways required = (

^{n}C

_{r}*

^{n}C

_{r}). Where

^{n}C

_{r}= n! / r!(n – r)!.

(

^{4}C

_{1}*

^{8}C

_{2}) = (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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