Question

There are three women and five men who will split up into two four-person teams. how many ways are there to do this so that there is (at least) one woman on each team?

Answer 1

Once the first team is defined, so is the other one. With no loss of generality, we can define the first team as the one with one (1) woman. There are 3 ways to have 1 woman on the team.

The remaining 3 team members are men. There are C(5,3) = 10 ways to choose 3 men from 5. Thus, the first team can be formed in 3×10 = 30 different ways.

There are 30 ways to form the teams as described.

_____

The function C(n, k) or nCk (choose k from a pool of n) is defined as

... nCk = n!/(k!(n-k)!)

so 5C3 = 5!/(3!(5-3)!) = 5·4/2 = 10