Question

A committee of 5 is to be formed from a group of 10 people consisting of 4 single men, 4 single women and a married couple. The committee is to consist of a chairman, who must be a single man , two other men, and two women.
(i) Find total number of committees formed.
(ii) How many of these would include the married couple ?

Answer 1

Given group contains 10 people, in which 4 single men, 4 single women and a married couple.
To form a committee is to consist of a chairman, who must be a single man , two other men, and two women.
There are ${}^4{C}_1$ ways to chose the chairman.
There are ${}^4{C}_2$ ways of choosing the men which is 6
there are ${}^5{C}_2$ ways of choosing the women
Thus, the number of possible ways to form a committee ${=}^4{C}_1{\times }^4{C}_2{\times }^5{C}_2$
$4\times 6\times 10=240$ ways.

(ii) Now committee must contain married couple, so remaining three members of the team can be selected as follows.
the chair man can be chosen from 4 men as ${}^4{C}_1$ ways
there are 3 men left from the men. You get to chose 1 more because the chairman (already chosen) is a man and one of the men is married and he's on the committee, this can be done by ${}^3{C}_1$ ways.
One more woman is needed. There are four women left.
So this can be done by ${}^4{C}_1$ ways.
Therefore, the number of ways to form a committee included by married couple is ${}^4{C}_1{\times }^3{C}_1{\times }^4{C}_1=4\times 3\times 4=48$ ways.