This is a formula about the binomial.I am so confused to prove it by combination way.
$${\sum _{r=0}^{k}{\binom {n+r-1}{r}}={\binom {n+k}{k}}}$$ Formula
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You mean, how to prove it combinatorially? (Instead of say, algebraically) – M. Vinay Mar 18 '19 at 01:26
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@JMoravitz Is it exactly the same as the other one? Here $r$ in $\binom{n + r - 1}{r}$ varies, but in the other one, $k$ in $\binom t k$ does not. – M. Vinay Mar 18 '19 at 01:31
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@M.Vinay Yes, it is. The hockey stick identity can be written such that it travels diagonally down to the left ending with a sharp right turn along Paschal's triangle or as is the case here, diagonally down to the right with a sharp left at the end. The only steps to transform from the one version to the other is to remember that $\binom{t}{k} = \binom{t}{t-k}$ and re-index. This question was also closed as duplicate of the one I linked already for same reason. – JMoravitz Mar 18 '19 at 01:34
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@JMoravitz Thanks for clearing that up. – M. Vinay Mar 18 '19 at 01:35