ILL 1986 Problem 53

For given positive integers $r, v, n$ let $S(r, v, n)$ denote the number of $n$ -tuples of non-negative integers $(x_1, \cdots, x_n)$ satisfying the equation $x_1 +\cdots+ x_n = r$ and such that $x_i \leq v$ for $i = 1, \cdots , n$ . Prove that $$S(r, v, n)=\sum_{k=0}^{m} (-1)^k \binom nk \binom{r - (v + 1)k + n - 1}{n-1}$$ Where $m=\left\{n,\left[\frac{r}{v+1}\right]\right\}.$