Let S={1,2,3,4,5,6}. How many functions f(x) are there from S to S such that f(f(x))=x for all x ∈ S?

There are 6! functions that take S to itself, as a permutation.  Since we are looking for functions such that f(f(x)) = x, we must divide by 2, to get 6!/2 = 360.