Pascal's rule

Statement

Let $n$ and $k$ be natural numbers. Then $$\binom{n+1}{k+1} = \binom{n}{k} + \binom{n}{k+1}.$$

Remarks

For $n = 4$ and $k = 1$ it gives $\binom{5}{2} = \binom{4}{1} + \binom{4}{2} = 4 + 6 = 10$. Combinatorial proof: fix one element $x$ of an $(n+1)$-element set. A $(k+1)$-subset contains $x$, and then it has $k$ other elements, or it omits $x$. The rule builds Pascal's triangle row by row.

Explanations

L1 human

Each number in Pascal's triangle is the sum of the two numbers above it. This rule is the same fact: a team of $k+1$ from $n+1$ people either includes a fixed person or does not.