I have a test coming up tomorrow, and on one of the practice papers, this question showed up:
Show that for all $k \in \mathbb{N}$, and all $n \in \mathbb{N}\cup\{0\}$, $$\sum_{j=k}^{n+k}{j\choose k} = {n+k+1\choose k+1}$$
I don't know how to prove this at all. I know how to prove thing using mathematical induction, but it gets harder I find when the binomial theorem is involved, let alone multiple variables. Does anyone know how I'd go about answering this? The solutions weren't posted (and it's not a part of any assignment, so I'm just doing this for practice).