The Captain Cool

There are 6 girls and 9 boys.

If he selects 4 girls, he has 6!/((4!)((6-4)!)) = 6!((4!)(2!)) = 15 ways. Now he has to select 7 boys. He has 9!/((7!)((9-7)!)) = 9!((7!)(2!)) = 36 ways This give us 15*36 ways.

If he selects 5 girls, he has 6 ways to pick the girl who is not chosen, and therefore 6 ways to pick the girl that is chosen. Now he has to select 6 boys. He has 9!/((6!)((9-6)!)) = 9!((6!)(3!)) = 84 ways This give us 6*84 ways.

If he selects all 6 girls, he has only 1 way to do this. Now he has to select 5 boys. He has 9!/((5!)((9-5)!)) = 9!((5!)(4!)) = 126 ways This give us 1*126 ways.

Adding these all up give us 1170.

Out of the 15 selected initially 6 are girls while 9 are boys. To select 11 out of those 15 we can either choose 4 girls and 7 boys, 5 girls and 6 boys or 6 girls and 5 boys. The number of possible ways then can be calculated by: N= (6C4 * 9C7) + (6C5 * 9C6) + (6C6 * 9C5) N= 1170 Note that if two events must take place together like if we select 4 girls there have to be 7 boys the combinations will be multiplied and if two or more events can take place but not at the same time, we add. For example you can have 4 or 5 or 6 girls but not 4 and 5 and 6 girls together in a selection.