Number of Permutations of One Less/Proof 2

Theorem

${}^{n - 1} P_n = {}^n P_n$

where ${}^k P_n$ denotes the number of ordered selections of $k$ objects from $n$.

Proof

\(\ds {}^{n - 1} P_n\)

\(=\)

\(\ds n^{\underline {n - 1} }\)

Number of Permutations: $n^{\underline {n - 1} }$ denotes Falling Factorial

\(\ds \)

\(=\)

\(\ds n!\)

Integer to Power of Itself Less One Falling is Factorial

\(\ds \)

\(=\)

\(\ds {}^n P_n\)

Number of Permutations

$\blacksquare$