Symmetry of binomial coefficients

Statement

Let $n$ and $k$ be natural numbers with $k \le n$. Then $$\binom{n}{k} = \binom{n}{n - k}.$$

Remarks

For a set $X$ with $n$ elements, the map $S \mapsto X \setminus S$ is a bijection from the $k$-subsets of $X$ to the $(n-k)$-subsets. This is a bijective proof of the identity. The identity makes each row of Pascal's triangle read the same from left to right and from right to left.

Explanations

L1 human

To choose $k$ people for a team is the same as to choose the $n - k$ people who stay out. For example, $\binom{5}{2} = \binom{5}{3} = 10$.