Question

(vi) A club has 5 girls and 7 boys If 4 persons out of these are to be selected, find the total number of choices if : (a) there is no restriction on gender (b) 3 boys and 1 girl is to be selected​

Answer 1

Given,

Number of girls in club = 5

Number of boys in club = 7

4 members out of these are selected

To find,

(a) Ways of selecting 4 members without gender restriction

(b) Ways of selecting 3 boys and 1 girl

Solution,

Total number of members in the club = 5 girls + 7 boys = 12

We can use the permutations and combinations concept to solve this problem.

(a) Ways of selecting 4 members without gender restriction

We have to randomly select 4 members from the total members without any restriction.

Ways of selecting 4 members out of 12 members = $\ 12C_4$

$\ 12C_4 =$   $\frac{12!}{4!(12-4)!}$

= $\frac{12!}{4!(8)!}$

= $\frac{12*11*10*9}{4*3*2*1}$

= 11 x 5 x 9

= 495 ways

(b) Ways of selecting 3 boys and 1 girl

3 boys should be selected from 7, which is $^7C_3$ = 35

1 girl should be selected from 5, which is $^5C_1$ = 5

Total ways = $^7C_3$ + $^5C_1$ = 35+5 = 40

Hence, ways of selecting 4 members without gender restriction is 495 and  ways of selecting 3 boys and 1 girl is 40.