The number of $n$-tuples of nonnegative integers $(x_1,ldots,x_n)$ satisfying the equation $x_1+cdots+x_n = r$

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

If $v \geq r$, then we see that the sum is just $\binom{n+r-1}{n-1}$, which is the number of nonnegative integer solutions to $x_1+\cdots+x_n = r$. If $v < r$, then how do we get the formula? How is $v+1$ involved?