Exercise 1549Combinatorics › Binomial coefficientsOlympiadAdded 3 Nov 2026

Quadratic coefficient sum

#quadratic#coefficient#binomial

Statement

Show that $\sum_{k=0}^{n}\binom{n}{k}\binom{k}{2}=n(n-1)2^{n-3}$ .

Hint

Use $k(k-1)\binom{n}{k}=n(n-1)\binom{n-2}{k-2}$ .

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Identity. $k(k-1)\binom{n}{k}=n(n-1)\binom{n-2}{k-2}$ .

Since $\binom{k}{2}=\frac{k(k-1)}{2}$ . $\binom{n}{k}\binom{k}{2}=\frac{n(n-1)}{2}\binom{n-2}{k-2}$ .

Sum. $\frac{n(n-1)}{2}\sum_j\binom{n-2}{j}=\frac{n(n-1)}{2}\cdot 2^{n-2}=n(n-1)2^{n-3}$ .

Conclusion. The identity holds.