---
tag: 0YRG241HQQ
kind: statement
role: theorem
name: Pascal's rule
msc: [05A10]
uses: [092GEF38YD]
sources:
  - url: https://en.wikipedia.org/wiki/Pascal's_rule
    locator: lead section
same_as:
  wikidata: Q17099544
  wikipedia: 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|05DME4TZ3K]] 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.