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
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.5$