Number of Permutations of One Less/Proof 1

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

$\blacksquare$