TITLE: Number of subsets with even number of elements
QUESTION [8 upvotes]: Let $|X|=n$. How to find all number of subsets $X$ consisting of an even number of elements?

REPLY [6 votes]: In total there are $2^n$ subsets of $X$.
If $n$ is odd then there is a one-to-one correspondence between sets with even cardinality and sets with odd cardinality. Subset $S$ corresponds with its complement. Consequently the number of subsets with even cardinality equals the number of subsets with odd cardinality. So this number is $\frac122^n=2^{n-1}$.
If $n$ is even then put one element $x_0\in X$ aside. With the trick described above we find that $X-\{x_0\}$ has $2^{n-2}$ subsets with even cardinality and also $2^{n-2}$ subsets with odd cardinality. The subsets of $X-\{x_0\}$ with odd cardinality become subsets $X$ with even cardinality if element $x_0$ is added to each of them. This gives us $2^{n-2}+2^{n-2}=2^{n-1}$ subsets of $X$ with even cardinality.
So in both cases the answer is $2^{n-1}$.