TITLE: Calculating number of lines and planes in affine space over finite field
QUESTION [8 upvotes]: Consider a finite field $F_p$ with $p$ elements; how would one calculate the number of lines and the number of planes in the affine space $F^3_p$?
If one knew the number of lines through a particular point, the number of planes could then be calculated by multiplying the number of lines through the origin by the number of points on any line.

REPLY [12 votes]: Let us solve the more general problem of finding the total number of $k$-dimensional affine subspaces of $\mathbb{F}_q^n$. 
The starting point is the easy to prove fact that the number of $k$-dimensional (vector) subspaces of a the vector space $\mathbb{F}_q^n$ is the Gaussian coefficient $${n \brack k}_q = \frac{(q^n - 1)(q^n - q)\cdots(q^n - q^{k-1})}{(q^k - 1)(q^k - q)\cdots(q^k - q^{k-1})}.$$
Now the $k$-dimensional affine subspaces are precisely the cosets of the form $x + U$ where $x \in \mathbb{F}_q^n$ and $U$ is a $k$-dimensional subspace of $\mathbb{F}_q^n$. But $x + U = y + U$ if and only if $x - y \in U$. Therefore, as $x$ goes through all the $q^n$ elements of $\mathbb{F}_q^n$, each coset $k$-dimensional coset corresponding to the vector subspace $U$ is counted $q^k$ times. And thus the total number of $k$-dimensional affine subspaces is equal to $$q^{n - k}{n \brack k}_q.$$