Supposed that a department contains 10 men and 15 women. How many ways are there to form a committee with six members if it must have more women than men?

1 Answer
Feb 27, 2016

There are 9646096460 ways to form the committee.

Explanation:

First we ask, how many ways are there to choose rr objects out of nn distinct objects?

The answer turns out to be

((n),(r)) = frac{n!}{r!(n-r)!}(nr)=n!r!(nr)!

So for example, how many ways are there to choose 2 men out of 10 men?

The answer is

((10),(2)) = frac{10!}{2! * 8!} = 45

Now, with the condition that there must be more women than men, we are left with only 3 options: 4w+2m, 5w+1m and 6w+0m.

Number of ways with 4 women and 2 men

((15),(4)) * ((10),(2)) = frac{15!}{4! * 11!} * frac{10!}{2! * 8!} = 1365xx45

= 61425

Number of ways with 5 women and 1 man

((15),(5)) * ((10),(1)) = frac{15!}{5! * 10!} * frac{10!}{1! * 9!} = 3003xx10

= 30030

Number of ways with 6 women and 0 man

((15),(6)) * ((10),(0)) = frac{15!}{6! * 9!} * frac{10!}{0! * 10!} = 5005xx1

= 5005

Now all you have to do is to add the 3 cases up.

61425 + 30030 + 5005 = 96460