Guys, I'm trying to prove the hockey-stick identity using a combinatoric proof, here's what I tried:[math]\sum ^{r}_{k=0}\binom{n+k}{k}= \binom{n+r+1}{r}[/math]first I turned the RHS into [imath]\binom{n}{0}+ \binom{n+1}{1}+\binom{n+2}{2}+\cdots + \binom{n+r}{r}[/imath] because I thought this might be easier to work with.
RHS: We have a group of n+r+1 and we want to choose r people to form a committee, this can be done in [imath]\binom{n+r+1}{r}[/imath] ways.
I'm stuck in the LHS, can someone please help?
If possible can someone give some general tips on how to approach these kinds of questions, for instance how to choose a counting problem that is related to a certain problem that I'm trying to solve.
RHS: We have a group of n+r+1 and we want to choose r people to form a committee, this can be done in [imath]\binom{n+r+1}{r}[/imath] ways.
I'm stuck in the LHS, can someone please help?
If possible can someone give some general tips on how to approach these kinds of questions, for instance how to choose a counting problem that is related to a certain problem that I'm trying to solve.