Factorial formula for binomial coefficients

Statement

Let $n$ and $k$ be natural numbers with $k \le n$. Then $$\binom{n}{k} = \frac{n!}{k!\,(n - k)!}.$$

Remarks

Equivalently, $k!\,(n-k)!\,\binom{n}{k} = n!$. In this form the identity has no division. The formula needs $k \le n$. For $k > n$ the left side is $0$ and $(n - k)!$ has no meaning in the natural numbers. A cancelled form is $\binom{n}{k} = \frac{(n)_k}{k!}$, with the falling factorial $(n)_k$.

Explanations

L1 human

Arrange all $n$ objects in a row in $n!$ ways, and take the first $k$ as the chosen set. Each choice of set appears $k!\,(n-k)!$ times, because the first $k$ and the last $n-k$ objects can be reordered freely.