Proof of the symmetry of binomial coefficients from the factorial formula

Proof

We compute both sides with the factorial formula.

1. Let $k \le n$. By the factorial formula, $\binom{n}{k} = \dfrac{n!}{k!\,(n-k)!}$.
2. The number $n - k$ is a natural number with $n - k \le n$. So the factorial formula also applies to $\binom{n}{n-k}$.
3. Since $n - (n - k) = k$, it gives $\binom{n}{n-k} = \dfrac{n!}{(n-k)!\,k!}$.
4. The denominators in steps 1 and 3 are equal, because multiplication is commutative. Hence $\binom{n}{k} = \binom{n}{n-k}$.