$$\binom{r}{r}+\binom{r+1}{r}+\binom{r+2}{r}+...+\binom{n}{r}=\binom{n+1}{r+1}$$
I know how to prove this by induction.
But I got stuck with proving it combinatorially.
How do I start? and please give some directions.
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Search the site for *hockey stick identity*. Most proofs probably use induction, but there may be combinatorial arguments also. – Jyrki Lahtonen Jan 23 '17 at 06:44
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I actually came up with a combinatorial way of proving this. But that solution has already appeared [in this answer](http://math.stackexchange.com/a/1491698/11619), so I won't waste bandwidth by reposting it. – Jyrki Lahtonen Jan 23 '17 at 06:49
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Actually scrolling further down the list of answers gives you [this](http://math.stackexchange.com/a/1490965/11619). The same idea but more verbose. – Jyrki Lahtonen Jan 23 '17 at 06:55