Alternating sum of binomial coefficients

Statement

Let $n$ be a positive integer. Then $$\sum_{k=0}^{n} (-1)^k \binom{n}{k} = 0.$$

Remarks

For $n = 0$ the sum is $1$, so the hypothesis $n \ge 1$ is necessary. For $n = 4$: $1 - 4 + 6 - 4 + 1 = 0$. The identity is the binomial theorem with $x = -1$ and $y = 1$. It says that a nonempty finite set has as many subsets of even size as of odd size.

Explanations

L1 human

Add and subtract the numbers of a row of Pascal's triangle in turn, and you get $0$. For a nonempty set, add or remove one fixed element. This pairs the even subsets with the odd subsets.