Summation by k of Product by r of x plus k minus r over Product by r less k of k minus r

Theorem

 * $\displaystyle \sum_{k \mathop = 1}^n \left({\dfrac {\displaystyle \prod_{\substack {1 \mathop \le r \mathop \le n \\ r \mathop \ne m} } \left({x + k - r}\right)} {\displaystyle \prod_{\substack {1 \mathop \le r \mathop \le n \\ r \mathop \ne k} } \left({k - r}\right)} }\right) = 1$

where $1 \le m \le n$ and $x$ is arbitrary.