When trying to prove a problem I find that I need the above formula to be true, but I have no idea how to prove it. I am trying to prove that a given probability mass function is equivalent to a hypergeometric distribution and this identity pops up.
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5Search for "Vandermonde convolution". Actually, now that I have written this comment, the appropriate links might pop up on the sidebar. – darij grinberg Oct 01 '19 at 21:26
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The combinatoric proof is that you have $a+b$ things and are counting the ways to choose $c$ of them. That is directly the interpretation of the right hand side. The left assumes you break the $a+b$ into a group of $a$ and a group of $b$, then choose $x$ of the $a$ and $c-x$ of the $b$. When you sum over $x$ from $0$ to $c$ you get all the possibilities.
Ross Millikan
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This has been asked and answered before (with combinatorial and other proofs). – Martin R Oct 02 '19 at 05:21