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 |
$\blacksquare$
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