Prove that $\sum _{k=\rho}^{n} (-1)^k \binom{k}{\rho} \binom{n}{k} = 0$

Question

I would like to prove that

$$\sum _{k=\rho}^{n} (-1)^k \binom{k}{\rho} \binom{n}{k} = 0 \ \ \ \forall n,\rho \in \mathbb{N}$$

I tested this numerically and it seems to hold, also it is needed for a bigger proof on which I am working at the moment. I tried by induction and by contradiction, with no luck. I also scouted the literature, with the same luck.