Exercise 806Combinatorics › Binomial coefficientsL1Added 3 Nov 2026

An inequality with binomials

#inequality#binomial#bound

Statement

Show that $\binom{n}{k}\le 2^n$ for all $0\le k\le n$ .

Hint

$\binom{n}{k}$ is one term of the sum $\sum_k\binom{n}{k}=2^n$ .

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Nonnegativity. All terms are nonnegative, so any single term is $\le$ the sum.

Conclusion. $\binom{n}{k}\le 2^n$ .

Refinement. $\binom{n}{k}\le\binom{n}{\lfloor n/2\rfloor}\approx\frac{2^n}{\sqrt{\pi n/2}}$ .