2019 Mock AMC 10B Problems/Problem 13

To start with, we can and should find the total number of ways Ms. Jannesen can choose pairs. We can imagine each student as a dot, and a pair of lines/borders to represent different groups. |..|..|..|..|..|..|..|..|..|..|..|..|..|..|..|..| Secondly, we can take the factorial of 32, since in doing so we can see how many ways we can rearrange the kids. And as we rearrange the kids, the grouping of the kids changes according to the diagram above. But we must consider that the order of the groups or the people in the group doesn't matter. In order to not overcount, we must divide