# use combinatorial reasoning to calculate $\sum{\binom{100}{a}\binom{200}{b}\binom{300}{c}}$

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.