I tried counting subsets with r+1 elements from the set {1,2,....n, n+1} which gives RHS. However I could not come up with an answer corresponding to LHS.
$$\binom{r}{r}+\binom{r+1}{r}+...+\binom{n}{r}=\binom{n+1}{r+1} $$
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Hint: consider the lexicographically largest of the things you wind up choosing from $n+1$. Where can it be, and how many ways are there to choose the rest of the things? – Steven Stadnicki Apr 13 '21 at 21:59
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1There are multiple proofs of several kinds at the link; the one by hunter some distance down is combinatorial and very clearly presented. – Brian M. Scott Apr 13 '21 at 22:22
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@BrianM.Scott Thank you for your answer. Had I known the duplicate answer, I would not have asked the question. I am new here, so should I delete this question? – Drilon Aliu Apr 13 '21 at 22:46
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@AlexParker: You’re welcome. It’s entirely up to you; someone who happens to stumble on this question gets a pointer to lots of answers, which is nice, but apart from that nothing is lost if you do delete it. – Brian M. Scott Apr 13 '21 at 22:58