How to prove combinatorial $\sum_{k=0}^{n/2} {n\choose2k} = \sum_{k=0}^{n/2 - 1} {n\choose2k+1} = 2^{n-1}$ I have problems solving the following formula for even positive integers $n$:
$$\sum_{k=0}^{n/2} {n\choose 2k} = \sum_{k=0}^{n/2 - 1} {n\choose 2k+1} = 2^{n-1}$$
I tried to prove it by induction but it didn't work. Is there any combinatorial proof?

Hint: 
You can get the result by expanding these two expressions using the binomial theorem
$$(1-1)^n=0\ \ \ \text{ and } \ \ \ (1+1)^n=2^n.$$
