Consider a set with $n$ elements.

How many subsets of odd cardinality exist?
Let $I_{n}$ denote the number of subsets of odd cardinality and $P_{n}$ the number of subsets of even cardinality.

We already know that

$$
I_{n}+P_{n}=2^{n}
$$

Let's show that $I_{n}=P_{n}$.

Using the binomial formula

We have already seen that

$$
(1-1)^{n}=\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}=\sum_{\substack{k \in\{1, \ldots, n\} \\ k \text { even }}}\binom{n}{k}-\sum_{\substack{k \in\{1, \ldots, n\} \\ k \text { odd }}}\binom{n}{k}
$$

which means

$$
0=P_{n}-I_{n}
$$

Regardless of the method, we conclude that

$$
\left\{\begin{array}{l}
I_{n}+P_{n}=2^{n} \\
I_{n}-P_{n}=0
\end{array}\right.
$$

and thus

$$
I_{n}=2^{n-1} \text {. }
$$