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.

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Tag	`092GEF38YD`
URI	`m2://092GEF38YD`
Kind	definition
Writing	W1 stated
MSC	05A10
Defines	Binomial coefficient

Binomial coefficient

Definition

Remarks

Where this leads

Further reading

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