Exercise 217Combinatorics › Binomial coefficientsHigh schoolAdded 3 Nov 2026

Symmetry of binomial coefficients

#symmetry#binomial coefficient#identity

Statement

Show that $\binom{n}{k}=\binom{n}{n-k}$ for $0\le k\le n$ .

Hint

Use the factorial formula, or a bijection (complement of a subset).

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Formula.
$\binom{n}{n-k}=\frac{n!}{(n-k)!\,(n-(n-k))!}=\frac{n!}{(n-k)!\,k!}=\binom{n}{k}.$

Bijection. Map a $k$ -subset to its complement (an $(n-k)$ -subset), which is a bijection.

Conclusion. $\binom{n}{k}=\binom{n}{n-k}$ .