I Number of Binary Operations on a Set with a Special Property

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TL;DR Summary

I was wondering if anyone could look over my solution to the question

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

Hello all,

The question I am tackling is as follows:

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

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

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.

Any feedback would be great!

 

[andrewkirk]

Very good. makes sense to me.

 

[Bosko]

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

Yes, you are right. $n^{n(n-1)}$ is the solution , if there no any other constraint on the binary operation *.
There are n(n-1) places in the matrix that are not on the diagonal.
On any of them you can put any of n elements of the set S.