Suppose that a department contains 14 men and 17 women How m

Suppose that a department contains 14 men and 17 women. How many ways are there to form a committee with 6 members if it must have strictly more women than men?

Solution

Three cases in which this can be done:

Case 1: 4 women ,2 men

This is done in:C(17,4)C(14,2) =216580 ways

Case 2: 5 women, 1 men

This is done in: C(17,5)C(14,1) =86632 ways

Case 3: 6 women ,0 men

This is done in: C(17,6)=12376 ways

So total number of ways is: 315588