1
$\begingroup$

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.

$\endgroup$
5
$\begingroup$

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.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.