There are ways to choose two elements for the first 2-cycle. There are ways to choose two elements for the second 2-cycle. Since you can choose the 2-cycles in either order, this double counts:
The n(n - 1)(n - 2)(n - 3) portion is trivial. Obviously this over counts the number of permutations we're looking for and it is natural to find a counting scheme to divide by to get the number of distinct 2-cycles, but I am at a loss to explain how to get the 8.Show that if then the number of permutations in S_n which are the product of two disjoint 2-cycles is n(n - 1)(n - 2)(n - 3)/8.