Proof that $C(n , 0) + C(n+1 , 1) + C(n+2 , 2) + ... + C(n + k , k) =C(n+k+1 , k)$

Question

Hello everyone I am a high school student and I need help with the proof that $$C(n , 0) + C(n+1 , 1) + C(n+2 , 2) + ... + C(n + k , k) =C(n+k+1 , k).$$

Answer 1

$$\sum_{m=0}^{k}\binom{n+m}{m}=\sum_{m=0}^{k}\binom{n+m}{n}$$ Setting $n+m \mapsto m$ and using Hockey-stick identity follows:

$$=\sum_{m=n}^{k+n}\binom{m}{n}=\binom{k+n+1}{n+1}=\binom{k+n+1}{k}$$

Which is the claim.