USAMO 1994 Problem 5

$\mathrm{X}$ is a set of $\mathrm{n}$ positive integers with sum $\mathrm{s}$ and product $\mathrm{p}$. Show for any integer $\mathrm{N}>=\mathrm{s}, \Sigma$ ( parity( $\mathrm{Y}$ ) ( $\mathrm{N}-$ sum(Y))Cs $)=\mathrm{p}$, where $\mathrm{aCb}$ is the binomial coefficient $\mathrm{a}!/(\mathrm{b}!(\mathrm{a}-\mathrm{b})$ ! ), the sum is taken over all subsets $\mathrm{Y}$ of $\mathrm{X}$, parity $(\mathrm{Y})=1$ if $\mathrm{Y}$ is empty or has an even number of elements, -1 if $\mathrm{Y}$ has an odd number of elements, and $\operatorname{sum}(\mathrm{Y})$ is the sum of the elements in $\mathrm{Y}$.