Find the sum of $$S=\binom{2007}{0}+\binom{2007}{4}+\binom{2007}{8}+...+\binom{2007}{2004}$$
My work so far: $$(1+1)^n=2^n=\binom{n}{0}+\binom{n}{1}+\binom{n}{2}+...+\binom{n}{n}$$ $$(1-1)^n=0=\binom{n}{0}-\binom{n}{1}+\binom{n}{2}-...+(-1)^n\binom{n}{n}$$