Factorial formula for binomial coefficients

Builds on Factorial Binomial coefficient

Statement

Let $n$ and $k$ be natural numbers with $k \le n$ . Then $\binom{n}{k} = \frac{n!}{k!\,(n - k)!}.$

Remarks

Equivalently, $k!\,(n-k)!\,\binom{n}{k} = n!$ . In this form the identity has no division. The formula needs $k \le n$ . For $k > n$ the left side is $0$ and $(n - k)!$ has no meaning in the natural numbers. A cancelled form is $\binom{n}{k} = \frac{(n)_k}{k!}$ , with the falling factorial $(n)_k$ .

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Binomial coefficient definition
Factorial definition
Factorial formula for binomial coefficients statement

Tag	`033KTKKRV6`
URI	`m2://033KTKKRV6`
Kind	statement, theorem
Writing	W0 stub
MSC	05A10
Status	This copy of the site has no build data. Run `m2 status 033KTKKRV6`.
Uses	Factorial, Binomial coefficient

Factorial formula for binomial coefficients

Statement

Remarks

Reading path

Further reading

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