Exercise 1545Combinatorics › Binomial coefficientsOlympiadAdded 3 Nov 2026

Partial sums of binomials

#partial sum#binomial#hockey stick

Statement

Compute $\sum_{k=0}^{n}\binom{k}{2}$ .

Hint

Hockey-stick with $r=2$ : $\sum_{k=2}^{n}\binom{k}{2}=\binom{n+1}{3}$ .

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Hockey-stick. $\sum_{k=2}^{n}\binom{k}{2}=\binom{n+1}{3}$ .

Check. $n=3$ : $\binom{2}{2}+\binom{3}{2}=1+3=4$ , and $\binom{4}{3}=4$ . ✓

Conclusion. $\sum_{k=0}^{n}\binom{k}{2}=\binom{n+1}{3}$ .