October 26, 2010

# Men and Women

**Problem**

How many different couples we can make if we are asked to select 5 women from 10 candidates and 5 men from 12 candidates.

**Solution**

There are 10 choose 5 different ways to select the women:

10!/(10-5)!5! = 252

There are 12 choose 5 different ways to select the men:

12!/(12-5)!5! = 792

Total number of combinations:

252 x 792 = 199584

We are not done yet because we need to pair 5 men with 5 women so the number of combinations should be multiplied by 5! because the first man can choose any women from 5, the next man can choose any one of the remaining 4 and so on.

So the total number of ways:

199584 x 5! = 23950080