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### Problem 7

(*) How many different commutative binary operations can be defined on a set of 2 elements? on a set

of 3 elements? on a set of n elements?

**Solution**

A commutative binary operation on a set of n elements is determined by the elements on or above the main diagonal of its table. This is the number of elements on the diagonal, $n$, plus the number of elements above the diagonal, $\frac{n^{2}-n}{2}$ (the full table, minus the main diagonal, divided by 2). This simplifies to $\frac{n^{2} + n}{2}$. Each element has $n$ options, so there are $n^{\frac{n^{2} + n}{2}}$ possible commutative binary operations, $c$, on an $n$-element set. So then for

(1)