Exercise 410Combinatorics › Binomial coefficientsHigh schoolAdded 3 Nov 2026

Binomial as a quotient of factorials

#factorial#quotient#binomial

Statement

Show that $\binom{n}{k}=\frac{n(n-1)\cdots(n-k+1)}{k!}$ for $1\le k\le n$ .

Hint

$\binom{n}{k}=\frac{n!}{k!(n-k)!}=\frac{n(n-1)\cdots(n-k+1)}{k!}$ .

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Compute.
$\frac{n!}{(n-k)!}=n(n-1)\cdots(n-k+1).$

So $\binom{n}{k}=\frac{n(n-1)\cdots(n-k+1)}{k!}$ .

Conclusion. The formula holds.

Application. For small $k$ , this is often easier to compute. E.g. $\binom{n}{2}=\frac{n(n-1)}{2}$ .