Combinatorial proof of $\sum_{i=0}^n {{i}\choose{ k}} = {{n+1}\choose{ k+1}}$

It’s easy to prove $\sum_{i=0}^n {{i}\choose{ k}} = {{n+1}\choose{ k+1}}$ by induction with ${{n}\choose{ k}} = {{n-1}\choose{ k-1}} + {{n}\choose{ k-1}}$, but what is its combinatorial proof?

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