Give the combinatorial proof of the identity $\sum\limits_{i=0}^{n} \binom{k-1+i}{k-1} = \binom{n+k}{k}$
HINTS:
Consider a $k$-element subset of $[n+k]=\{1,\dots,n+k\}$; it has a maximum element, which can be anything from $k$ through $n+k$. How many such subsets are there with maximum element $k+i$ for $i=0,\dots,n$?
There are $\binom{k-1+i}{k-1}$ compositions of $k+i$ with $k$ terms. There are $\binom{n+k}k$ compositions of $n+k+1$ with $k+1$ terms.
Use $\binom{k}{r} = \binom{k-1}{r} + \binom {k-1}{r-1}$ repeatedly on the expansion of sum.