Number of Binary Operations on a Set with a Special Property

How many different binary operations on a set S with n elements have the property that for all x ∈ S, x * x = x ?

Solution:

Thinking of all the possible operations as entries on an n x n matrix, the entries x * x would lie on the diagonal of the matrix. The total number of entries in the matrix would be n^2, and, since the elements on the diagonal of the matrix (the elements x * x) have a pre-determined value (and there are n of these elements), the number of elements that we need to map would total n^2 - n.

So, when when we map n^2 - n elements to n elements, there will be n^(n^2 - n) total binary operations.

Expiring said:

I was wondering if any of you could look over my solution and tell me if my logic is correct.