1994 USAMO 第 5 题

Let $\, |U|, \, \sigma(U) \,$ and $\, \pi(U) \,$ denote the number of elements, the sum, and the product, respectively, of a finite set $\, U \,$ of positive integers. (If $\, U \,$ is the empty set, $\, |U| = 0, \, \sigma(U) = 0, \, \pi(U) = 1$.) Let $\, S \,$ be a finite set of positive integers. As usual, let $\, \binom{n}{k} \,$ denote $\, n! \over k! \, (n-k)!$. Prove that

$$\sum_{U \subseteq S} (-1)^{|U|} \binom{m - \sigma(U)}{|S|} = \pi(S)$$

for all integers $\, m \geq \sigma(S)$.

令 $\, |U|, \, \sigma(U) \,$ 和 $\, \pi(U) \,$ 分别表示正整数有限集 $\, U \,$ 的元素数量、和和乘积。 (如果 $\, U \,$ 是空集，则 $\, |U| = 0, \, \sigma(U) = 0, \, \pi(U) = 1$。)令 $\, S \,$ 为正整数的有限集。像往常一样，令 $\, \binom{n}{k} \,$ 表示 $\, n！ \超过k！ \, (n-k)!$。证明

$$\sum_{U \subseteq S} (-1)^{|U|} \binom{m - \sigma(U)}{|S|} = \pi(S)$$

对于所有整数 $\, m \geq \sigma(S)$。