The number of ways to have a sum of $n$ natural numbers equal to $k$

Question

Show that $\sum_{i=1}^{n}\binom{n}{i}\binom{k+1}{i-1}=\binom{n+k-1}{k}$, where $n$ and $k$ are fixed natural numbers. This expression is for the number of ways to have a sum of positive or equal to zero natural numbers $\sum^n_{i=1} a_i$ equal to $k$. First factor is the number of ways to pick the boxes where I put my '$1$'s, the second factor is the number of ways to have these $k$ '$1$'s put into the selected number of boxes (using the stars and bars technique). The right hand side is in the answer. I can't figure out the transition.

Answer 1

I think that the given expression which is being asked to prove is not correct.

for example put n=4 and k=2 the expression in LHS becomes 35 while RHS is only 10

so... please recheck the question. I'm writing this here as I don't have the required reputation to comment on your question.