Sum of sequence of $\binom{n}{r}$ How can we find the sum of $ \binom{21}{1}+ 5\binom{21}{5}+ 9\binom{21}{9}....+17\binom{21}{17}+ 21\binom{21}{21}$?
I have no clue how to begin. I guess complex numbers might help. 
EDIT: Actually the real question was that the sum above was k and we had to find its prime factors. And the answer according to the key involves complex numbers. They have directly written that:
$k=\frac{21( 2^{20} + 0^{20} + (1+i)^{20} + (1-i)^{20})}{4} = 21(2^{18}-2^9)$
I didn't get the above.

An addenudm to ILUA's answer. General methods for this kind of identities are treated in the gorgeous generatingfunctionology by Herbert S. Wilf. For this particular method, see Example 4 at page 51, especially the remark at the end.
