Exercise 1613Combinatorics › Counting principlesOlympiadAdded 3 Nov 2026

A combinatorial identity via inclusion-exclusion

#inclusion-exclusion#binomial identity

Statement

Show $\sum_{k=0}^{n}(-1)^k\binom{n}{k}=0$ for $n\ge1$ .

Hint

Expand $(1-1)^n$ or use inclusion-exclusion on empty subset counting.

Show solutionHide solution

Binomial. $(1-1)^n=\sum_{k}\binom{n}{k}(-1)^k=0$ .

Inclusion-exclusion interpretation. Number of subsets of an $n$ -set with no element = $0$ for $n\ge1$ .

Conclusion. The identity holds.