Any help is welcome! Also I'm searching for specifically combinatorial proof, not algebraic.
Edit: What I tried so far (for right side): $N$ is a set with $n$ elements, and one of the elements is fixed. Then the right side is number of ways to choose $k−1$-element subset of $n−1$-element set ($N$ without the fixed element), and then 'highlight' one element from that subset in union with the fixed element (so in total, highlight one of $(k−1)+1=k$ elements). I think this is wrong approach but I can't come up with anything else.
Best Answer
Count chaired committees of size $k$ chosen from among people $\{1,\dots,n\}$, where the chairperson must be chosen from among $\{1,\dots,k\}$. The RHS first selects the chairperson in $\binom{k}{1}$ ways and then chooses the remaining $k-1$ members from among the remaining $n-1$ people in $\binom{n-1}{k-1}$ ways. The LHS conditions on the number $i$ of members chosen from $\{1,\dots,k\}$. First choose these $i$ members in $\binom{k}{i}$ ways, then choose the chairperson from among these $i$ members in $\binom{i}{1}$ ways, and then choose the remaining $k-i$ members from among $\{k+1,\dots,n\}$ in $\binom{n-k}{k-i}$ ways.