How many different teams can be chosen from a group if each team must have at least two girls on it?

Question

How many different teams of 5 children can be chosen from a group of 18 girls and 7 boys if each team must have at least two girls on it? I know it's a combination of 18Cx*7Cy and x+y=5, but I don't know how to set this problem up.

Answer 1

Each team needs two girls, so we have a choice of 18. After picking one girl there are 17 left, so we pick one. It doesn't matter which order so, since there are 2 different ways of arranging 2 girls, the number of ways of combining 2 girls (18C2) is 18*17/2=153. There are 23 children left to choose from and we need 3 of these. There are 6 ways of arranging 3 different children in order, so simply combining them is given by 23*22*21/6=1771 (23C3). Combining the two results we get 153*1771=270,963.

Best to try and use simple logic. That's what mathematics is really all about. Formulas and prescriptions may speed things up but they don't necessarily help you understand what's going on.