Hall's identity

Hall's identity is a mathematical result related to the theory of partitions and combinatorial identities. Specifically, it provides a relationship involving binomial coefficients, which can be viewed through the lens of combinatorial enumeration. The identity states that for any non-negative integer \( n \): \[ \sum_{k=0}^{n} (-1)^k \binom{n}{k} (n-k)^m = (-1)^n \binom{m}{n} n!