Sum over k of r Choose k by s-kt Choose r by -1^k

Theorem
Let $r \in \Z_{\ge 0}$.

Then:
 * $\displaystyle \sum_k \binom r k \binom {s = k t} r \left({-1}\right)^k = t^r$

where $\dbinom r k$ etc. are binomial coefficients.

Proof
From the Sum over $k$ of $\dbinom r k \left({-1}\right)^k$ by Polynomial:


 * $\displaystyle \sum_k \binom r k \left({-1}\right)^{r - k} P_r \left({k}\right) = r! \, b_r$

where:
 * $P_r \left({k}\right) = b_0 + b_1 k + \cdots + b_r k^r$ is a polynomial in $k$ of degree $r$.