Exercise 1533Combinatorics › Binomial coefficientsOlympiadAdded 3 Nov 2026

A combinatorial sum with choice

#combinatorial sum#binomial#choice

Statement

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

Hint

Use $k\binom{n}{k}=n\binom{n-1}{k-1}$ and Vandermonde.

Show solutionHide solution

Rewrite. $k\binom{n}{k}=n\binom{n-1}{k-1}$ .

Sum. $\sum_k k\binom{n}{k}^2=n\sum_k\binom{n-1}{k-1}\binom{n}{k}$ .

Vandermonde variant. Using $\binom{n-1}{k-1}=\binom{n-1}{n-k}$ , $\sum_k\binom{n-1}{n-k}\binom{n}{k}=\binom{2n-1}{n}$ .

Conclusion. $\sum_k k\binom{n}{k}^2=n\binom{2n-1}{n}=\frac{n}{2}\binom{2n}{n}$ .