Number of Permutations of All Elements/Proof 1

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

$\blacksquare$


Sources

  • 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text I$. Algebra: Permutations and Combinations: The number of ways of arranging $n$ things in line
  • 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {3-1}$ Permutations and Combinations: Theorem $\text {3-1}$
  • 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials
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