Vandermonde's Identity: (1+x)^m*(1+x)^n = (1+x)^(m+n)

[Trenters4325]

To prove Vandermonde's identity

with (1+x)^m*(1+x)^n = (1+x)^(m+n),

you will get something like

$\displaystyle \sum^{m}_{a=0}\sum^{n}_{b=0}C(m,a)C(m,b)*x^{a+b} = \sum^{m+n}_{c=0}C(m+n,c)x^c$.

I see how you could set a+b = c, but I am having trouble rigorously justifying how you get rid of the summations?