Question

In how many ways can John and his 11 friends divide themselves into two teams each consisting of 4 people, two teams each consisting of 2 people?

Answer 1

There are 12 people.

First we select 4 out of this 12 people, then select 4 out of the remaining 8 people

But actually this is not the correct value. That is, we have double counted each selection. To explain this, let us denote the 12 people as P1, P2, P3, ..... P12, Now let us say in first selection we pick {P1, P2, P3, P4} and in second selection we pick {P5, P6, P7, P8}. We could also have selected {P5, P6, P7, P8} first and then {P1, P2, P3, P4} in second selection. From perspective of dividing into teams , it is identical. Hence we must divide by 2

Similarly, we can select the 2 teams of 2 each from the remaining 4 people in

Once we multiply these two ways, we get total ways