First, you need to put a [t e x] [/t e x] tag pair around the LaTex in the forum [without the spaces in the 't e x' of course]. However, that doesn't seem to take care of the space issue, i.e. the expression becomes
\(\displaystyle \sum_{k=0}^{500} {3000 -2k \choose 2000}{2001 \choose k}\)
So, formatting the LaTex a bit differently, I think you have the question:
How to calculate the sum
\(\displaystyle \sum_{k=0}^{500}\,\, \left( {\begin{array} {*{20}c} 3000-2k \\2000\end{array}} \right)\, \left( {\begin{array}{*{20}c} 2001 \\k\end{array}} \right)\)
using the method of generating functions?
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