Number of Total Orderings on Finite Set

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Theorem

Let $S$ be a finite set with $n$ elements.

Then there are $n!$ different total orderings that can be applied to $S$.


Proof

A total ordering on $S$ is by definition a permutation on $S$ in the sense of an ordered selection.

The result follows from Number of Permutations.

$\blacksquare$


Sources