How many committees of four could be created if. . .

Question

In a club with 8 men and 10 women, how many committees could be formed if at least 1 man and 1 woman must be on the committee?

Answer 1

The number of ways of choosing a committee of 4 out of 18 people is 18×17×16×15/4!=3060.

Of these there will be committees of 4 men and committees of 4 women. These must be excluded, leaving committees consisting of at least one man and one woman. 

To exclude the all men committee we must remove all combinations of 4 men chosen from 8=8×7×6×5/4!=70. To exclude the all women committee we must remove all combinations of 4 women chosen from 10=10×9×8×7/4!=210. So if we exclude all men and all women committees we get:

3060-(70+210)=2,780 committees including at least one man and one woman.