Question

A committee of 12 members is to be formed from 9 women and 8 men. In how many ways can this be done, if at least five women have to be included in a committee?

Answer 1

There are 9 women and 8 men. A committee of 12 members, consisting of at least 5 women can be formed by choosing,

5 women and 7 men; 6 women and 6 men;

7 women and 5 men; 8 women and 4 men;

9 women and 3 men;

$\therefore$ Total number of ways of forming the committee

${=}^9{C}_5{\times }^8{C}_7{+}^9{C}_6{+}^9{C}_7{\times }^8{C}_5{+}^9{C}_8{\times }^8{C}_4{+}^9{C}_9{\times }^8{C}_3$

= $126\times 8+84\times 28+36\times 56+9\times 70+1\times 56$

= $1008+2352+2016+630+56=6062$