Number of Permutations of One Less/Proof 1
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Theorem
- ${}^{n - 1} P_n = {}^n P_n$
where ${}^k P_n$ denotes the number of ordered selections of $k$ objects from $n$.
Proof
\(\ds {}^{n - 1} P_n\) | \(=\) | \(\ds \dfrac {n!} {\paren {n - \paren {n - 1} }!}\) | Number of Permutations | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {n!} {1!}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds n!\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds {}^n P_n\) | Number of Permutations |
$\blacksquare$