The sum of the product of two binomial coefficients

Question

This is really phsycics related question with mathematics behind it.

In my physics book there's the following relation:

$$\sum\binom{x_a+y_a-1}{x_a}\binom{x_b+y_b-1}{x_b}=\binom{x_a+x_b+y_a+y_b-1}{x_a+x_b}$$

For $x_a+x_b=c$ with $c$ some constant. (e.g. you can summarize when $x_a=1$ and $x_b=2$ for a total of $x_a+x_b=3$, but also $x_a=2$ and $x_b=1$, etc. any value from $x_a=0$ or $x_b=0$ is allowed. I'm sorry if this notation is not mathematical enough.)

I'm having some trouble deriving what's on the right hand side of the equals sign. If anyone could give some pointers or something that'd be nice. :-)

Answer 1

This is the Vandermonde identity. Proofs can be found here.