Prove that $\left(\!\!\binom{n\vphantom{1}}{k}\!\!\right)=\sum_{i=0}^k \left(\!\!\binom{n-1\vphantom{1}}{i}\!\!\right)$

Question

Let $n$ and $k$ be positive integers. Prove that: $$\left(\!\!\binom{n\vphantom{1}}{k}\!\!\right)=\sum_{i=0}^k\left( \binom{n-1\vphantom{1}}{i}\right)$$

Edit: This is asking for a direct proof. Sorry if it is unclear, it's exactly what was asked of me in the quiz and I want to be sure I get it right! Thanks

Answer 1

HINT: Since

$$\left(\!\!\binom{n}k\!\!\right)=\binom{n+k-1}{n-1}$$

by a stars and bars argument, it’s an immediate consequence of the hockey stick identity.