Given $ a + b + c = 100 $. $a,\ b,\ c $ are non-negative integers. Calculate $$ \sum {\binom{100}{a} \binom{200}{b} \binom{300}{c} } $$

Can someone help me with this question? I have no idea how to start it.


This sum is just $\binom{600}{100}.$

To see this, split a set of $600$ elements into three sets of sizes $300,200,$ and $100.$ Then to choose $100$ elements from the big set, we need to choose $a,b,c$ elements from those three sets for some nonnegative integers $a,b,c.$ Once we've decided how many elements we are choosing, we have $\binom{300}{a}\binom{200}{b}\binom{100}{c}$ ways to choose them.

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  • $\begingroup$ To add some reference, this is sometimes known as Vandermonde's convolution $\endgroup$ – Jean-Sébastien Oct 18 '13 at 3:10

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