Let $\Bbb{F}_2^n$ be $n$-dimensional vector space over $\Bbb{F}_2$, the two element field.
Approximately how many distinct subspaces $U\le \Bbb{F}_2^n$ of codimension 1 (i.e., dimension $n-1$) are there for large $n$? More generally, for fixed $k$, how many distinct subspaces $U\le \Bbb{F}_2^n$ of codimension $k$ (i.e., dimension $n-k$) are there for large $n$? Here, subspaces $U,U'$ are considered distinct if they have different sets of elements, (i.e., we don't care about their bases or anything).
The set of all hyperplanes is in bijection with the set of lines of the dual space, which is isomorphic to $\mathbb{F}_2^n$. So you have exactly $$2^n-1$$
hyperplanes in $\mathbb{F}_2^n$.