Exercise 810Combinatorics › Binomial coefficientsL1Added 3 Nov 2026

Generalized binomial coefficients

#generalized binomial#negative exponent#series

Statement

Define $\binom{-n}{k}$ for $n>0$ , $k\ge0$ , and compute $\binom{-3}{2}$ .

Hint

$\binom{-n}{k}=(-1)^k\binom{n+k-1}{k}$ .

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Definition. $\binom{-n}{k}=\frac{(-n)(-n-1)\cdots(-n-k+1)}{k!}=(-1)^k\binom{n+k-1}{k}$ .

Compute. $\binom{-3}{2}=(-1)^2\binom{4}{2}=6$ .

Application. $(1-x)^{-n}=\sum_{k\ge0}\binom{n+k-1}{k}x^k$ .

Conclusion. $\binom{-3}{2}=6$ .