How to prove this Binomial identity: $\binom{n}{k+1}=\sum_{i=0}^{n-k-1} \binom{k+i}{k}$

Question

I'm having problems solving this equation. I always get the wrong answer. Can somebody please help me solve it. Appreciate any help. I would appreciate the steps as well since I dont know where i messed up.

Thanks

Answer 1

Hint: In any selection of $k+1$ elements from $\{ 1, 2, \dots, n \}$, the greatest element selected is equal to $k+i+1$ for a unique value of $i$ in the range $0 \le i \le n-k-1$.

Answer 2

Use induction on $n$: suppose the formula is true for some $n$. Then \begin{align} \binom{n+1}{k+1}&=\binom{n}{k+1}+\binom{n}{k}&&\text{by Pascal's formula}\\ &=\sum_{i=0}^{n-k-1}\binom{k+i}{k}+\binom{k+(n-k)}{k} &&\text{by the inductive hypothesis}\\ &=\sum_{i=0}^{(n+1)-k-1}\binom{k+i}{k}. \end{align}