I know Pascal's Identity is ${n \choose k}={n-1 \choose k-1}{n-1 \choose k}$, but I am not sure how to set up and use the proof to show that $\sum_{k=0}^{r}C(n+k,k) = C(n+r+1,r)$.
Can anyone help me with this problem?
2026-04-03 04:21:05.1775190065
Use induction and Pascal's Identity to show that $\sum_{k=0}^{r}C(n+k,k) = C(n+r+1,r)$
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First verify $r=0$, then prove an induction on $r$ using Pascal's identity.