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:


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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