Number of Permutations of All Elements/Proof 1

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Theorem

Let $S$ be a set of $n$ elements.


The number of permutations of $S$ is $n!$


Proof

We are seeking to calculate the number of $r$-permutations of $S$, that is ${}^n P_r$, where $r = n$.

Hence:

\(\ds {}^n P_n\) \(=\) \(\ds \dfrac {n!} {\paren {n - n}!}\) Number of Permutations
\(\ds \) \(=\) \(\ds n!\) Definition of Factorial

$\blacksquare$


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