Number of closed binary operations which have an identity?

Question

Let $M=\{a_1,a_2,...,a_n\}$, $n\in \mathbb{N}$ be a set. What is the number of closed binary operations defined on $M$ which have an identity?
My approach : Let $a_1$ be the identity. Considering the Cayley Table of the binary operation, the column and the row of $a_1$ are perfectly determined since it is the identity.
We are left with counting the number of functions from a set with $n^2-2n+1$ elements to a set with $n$ elements. This is equal to $n^{n^2-2n+1}$.
The same reasoning works for all $a_i$, $i=\overline{1,n}$. As a result, the number of closed binary operations defined on $M$ which have an identity is $n^{n^2-2n+2}$.
I would like to know if my approach is correct.

Answer 1

Your answer is correct. The only way this could be wrong would be if there were binary operations which had two identities, because such operations would be counted twice by your method. However, it is well known that any operation has at most one (two-sided) identity.