Binomial coefficient

Definition

Let $n$ and $k$ be natural numbers. The binomial coefficient $\binom{n}{k}$, read "$n$ choose $k$", is the natural number defined by the recursion $$\binom{n}{0} = 1, \qquad \binom{0}{k+1} = 0, \qquad \binom{n+1}{k+1} = \binom{n}{k} + \binom{n}{k+1}.$$

Remarks

The value $\binom{n}{k}$ is the number of k-element subsets of an $n$-element set. Many texts take this as the definition. For $k > n$, $\binom{n}{k} = 0$. For $k \le n$, the factorial formula gives $\binom{n}{k} = \frac{n!}{k!\,(n-k)!}$. The recursion is Pascal's rule, and the values form Pascal's triangle.

Explanations

L1 human

The number $\binom{n}{k}$ tells how many ways there are to choose $k$ objects out of $n$ when the order does not matter. For example, $\binom{4}{2} = 6$: from $\{1, 2, 3, 4\}$ you can choose $12, 13, 14, 23, 24, 34$.