Question

From a group of 8 women and 6 men, a committee consisting of 3 men and 3 women is to be formed. How many different committees are possible if

(a) 2 of the men refuse to serve together?
(b) 2 of the women refuse to serve together?
(c) 1 man and 1 woman refuse to serve together?

Answer 1

Answer:

(a) 896

(b) 1000

(c) 910

Step-by-step explanation:

Number of women = 8

Number of men = 6

A committee consisting of 3 men and 3 women is to be formed.

(a) 2 of the men refuse to serve together.

Number of men who don't want to serve together = 2

Remaining men = 4

Total ways = Both men who don't want to serve together are not selected + only one of them is selected.

$^{8}C_{3}\cdot ^{4}C_{3}+^{8}C_{3}\cdot ^{4}C_{2}\cdot ^{2}C_{1}$

$896$

(b) 2 of the women refuse to serve together.

Number of women who don't want to serve together = 2

Remaining men = 6

Total ways = Both women who don't want to serve together are not selected + only one of them is selected.

$^{6}C_{3}\cdot ^{6}C_{3}+^{6}C_{3}\cdot ^{6}C_{2}\cdot ^{2}C_{1}$

$1000$

(c)1 man and 1 woman refuse to serve together.

Ways = Total ways - Both man and woman are select who refuse the serve together

$^{8}C_{3}\cdot ^{6}C_{3}-^{5}C_{2}\cdot ^{7}C_{2}$

$910$