Sophomore + Junior + Senior

Question

A class is attended by $n$ sophomores, $n$ juniors and $n$ seniors. In how many ways can these students form $n$ groups of three people each if each group is to contain a sophomore, a junior, and a senior?

Answer 1

First, line up the sophmores in any way. Next, line up the juniors across from the sophmores in any one of $n!$ ways. After that, line up the seniors across from the juniors in any one of $n!$ ways. The result is a total of $(n!)^2$ ways.

Answer 2

This is an exercise from A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory.

Hint: Simplify the problem by taking away the seniors. Can you solve the problem now? What if you add the seniors back in?