---
tag: 0PQQGFAWSH
kind: statement
role: theorem
name: Alternating sum of binomial coefficients
msc: [05A10]
uses: [092GEF38YD]
sources:
  - url: https://en.wikipedia.org/wiki/Binomial_coefficient
    locator: Sums of the 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|0EHWPWGHB2]] 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|05DME4TZ3K]] 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.