---
tag: 01WHD9T0V6
kind: definition
name: Central binomial coefficient
msc: [05A10]
defines: 0JPH2Y4DFV
sources:
  - url: https://en.wikipedia.org/wiki/Central_binomial_coefficient
    locator: lead section
---

## Definition

Let $n$ be a natural number.
The $n$-th central binomial coefficient is the [[binomial coefficient|092GEF38YD]]
$$\binom{2n}{n} = \frac{(2n)!}{(n!)^2}.$$

## Remarks

The first values, for $n = 0, 1, 2, 3, 4$, are $1, 2, 6, 20, 70$.
It is the middle entry of row $2n$ of [[Pascal's triangle|05DME4TZ3K]], and the largest entry of that row.
It counts the [[lattice paths|040WY0TT2G]] from $(0, 0)$ to $(n, n)$ with unit steps right and up. The [[closed formula for Catalan numbers|0NNDQVJ54Q]] divides it by $n + 1$.

## Explanations

### L1 human

The central binomial coefficient is the number in the middle of an even row of Pascal's triangle.
For $n = 2$ it is $\binom{4}{2} = 6$: the ways to choose 2 of 4 objects.