Subsets and their complements

Method 1. For $n\ge1$, subsets pair up $(S,\bar S)$ with $|S|+|\bar S|=n$. If $n$ is odd, exactly one is even; there are $2^{n-1}$ even subsets. If $n$ is even, the pairing preserves parity; a more refined argument shows exactly half are even: $2^{n-1}$.

Method 2. $\sum_{k\text{ even}}\binom{n}{k}=\frac{(1+1)^n+(1-1)^n}{2}=2^{n-1}$.

Conclusion. There are $2^{n-1}$ subsets of even size.