Question

A committee of $12$ is to be formed from nine women and eight men. In how many ways this be done if at least five women have to be included in a committee? In how many of these committees
a. the women hold majority?
b. the men hold majority?

Answer 1

A committee of 12 is to be formed from 9 women and 8 men with minimum 5 women. Then we have following selection ways.

Women (9)	Men (8)	Number of ways of selecting
5	7	${}^9{C}_5{\times }^8{C}_7=1008=s_1$
6	6	${}^9{C}_6{\times }^9{C}_6=2352=s_2$
7	5	${}^9{C}_7{\times }^8{C}_5=2016=s_3$
8	4	${}^9{C}_8{\times }^8{C}_4=630=s_4$
9	3	${}^9{C}_9{\times }^8{C}_3=56=s_5$
	Total	$=6062$

a. Number of committee when women are in majority is $s_3+s_4+s_5=2702$.
b. Number of committee when men are in majority is $s_1=1008$.