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. :-)

Summing over $x_a$ but $x_a$ appears in the RHS? What the... — Did, Aug 20 '14 at 12:36
@Did, that's because you are summing over $x_a+x_b=c$, and thus, in the RHS only constants appear — Ragnar, Aug 20 '14 at 12:40

score 1 · Accepted Answer · answered Aug 20 '14 at 12:33

1

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

answered Aug 20 '14 at 12:33

Batman

19,390

I don't understand though, Chu-Vandermonde identity applied correctly should give -2 in the top term of the binomial on the RHS no? – Felicio Grande Oct 10 '16 at 08:04

The sum of the product of two binomial coefficients

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