# Counting Possible Operations

Algebra Level 4

Let $$A$$ be any set:

An operation $$*$$ on $$A$$ is a rule which assigns to each ordered pair $$(a,b)$$ of elements of $$A$$ exactly one element $$a*b$$ in $$A.$$

If, for example, $$A$$ is a set consisting of just two distinct elements, say $$a$$ and $$b$$, each operation on $$A$$ may be described by a table such as the one below:

 $$(x,y)$$ $$x*y$$ $$(a,a)$$ $$(a,b)$$ $$(b,a)$$ $$(b,b)$$

Where $$x*y$$ could be either of the elements of $$A$$ ($$a$$ or $$b$$) for any $$(x,y)$$ in $$A$$. In general, there are many possible operations on a given set. A set containing just two elements for example, has $$16$$ possible operations.

How many possible operations are there on a set containing $$n$$ elements?

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