Does anyone know a combinatorial proof (or in general a proof, which doesn't work with induction) for the identity: $$ \sum_{i=1}^{n-1} \binom{m+i-1}{m} = \binom{m+n-1}{m+1} $$
With induction the proof is really easy, just using Pascal's Identity, but I would like to have a combinatorial proof (i.e. saying the LH side counts a set $Y$ and if we count $Y$ in a different way we get the RH).
This is called the Hockey Stick Identity (if you draw this on Pascal's triangle, it looks like a hockey stick).
For reference, usually the identity is written as $$\sum_{i=r}^n \binom{i}{r} = \binom{n+1}{r+1},$$ but this is equivalent to what you wrote (just pointing this out so it won't be confusing when you read other references). For combinatorial proofs, you can check out these references:
https://en.wikipedia.org/wiki/Hockey-stick_identity
https://artofproblemsolving.com/wiki/index.php/Combinatorial_identity