Combinatorics Confusion

Discrete Mathematics Level pending

\[ {\sum^{650} _{x+y=650}}\large \dbinom{2x}{x} \dbinom{2y}{y} \]

If \(x\) and \(y\) are non-negative integers, find S, if:
summation mod 7 = a;
summation mod 8 = b;
summation mod 9 = c;
summation mod 10 = d; and
S = a + b + c + d.

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