Cardinality of even and odd number of elements in Power Set

Question

I wish to prove that the number of sets in a power set is $2^n$ by using the fact that $$\sum^{n+1}_{k=0} \binom{n}{k} a^kb^{n+1-k}.$$ After that, I wish to use that sum formula to prove that the cardinality of even and odd number of elements in a power set have the same quantity, so it is always $2^{n-1}$, but how can I use the sum formula to prove that?

Answer 1

Use the binomial theorem with (i) a=1, b=1 (ii) a=1, b=-1