ThmDex – An index of mathematical definitions, results, and conjectures.

Let $n, m \in \mathbb{N}$ each be a D996: Natural number.

The first claim is clear, so assume that $m \leq n$. Using results

(i)	R5111: Number of injections is proportional to number of subsets of given size
(ii)	R1854: Cardinality of the set of injections between finite sets

we have \begin{equation} \begin{split} m ! \binom{n}{m} & = |\text{Inj}(\{ 1, \ldots, m \} \to \{ 1, \ldots, n \})| \\ & = \frac{n !}{(n - m) !} \end{split} \end{equation} Dividing both sides by the nonzero quantity $m !$, the claim follows. $\square$

(1)	\begin{equation} m > n \quad \implies \quad \binom{n}{m} = 0 \end{equation}
(2)	\begin{equation} m \leq n \quad \implies \quad \binom{n}{m} = \frac{n !}{(n - m) ! m !} \end{equation}