Exercise 400Combinatorics › Binomial coefficientsHigh schoolAdded 3 Nov 2026

Alternating sum of binomial coefficients

#alternating sum#binomial#zero

Statement

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

Hint

Set $a=1$ , $b=-1$ in the binomial theorem.

Show solutionHide solution

Binomial theorem. $(1-1)^n=\sum_k\binom{n}{k}(-1)^k=0$ for $n\ge1$ .

Conclusion. The alternating sum is zero.

Equivalent. The number of even-size subsets equals the number of odd-size subsets.