# Ordinary Lines in $\mathbb{R}^2$ and Other Fields

Suppose that we are given a bunch of points in $P \subseteq \mathbb{R}^2$which are not collinear. We say that a line is $k$-rich if it goes through $k$ points. If $k=2$, then we call the line ordinary. How would we determine the number of $k$-rich lines in $P$ if $k \geq 2$? Also what happens if we replace the parent field with $\mathbb{C}$ or another field? Do Hasse Diagrams come into play at all?

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One way is to choose every pair of points, compute the line between them, and see how many other points are on that line. If the answer is exactly $k$, increment our counter by one. When we are done, divide our counter by ${k\choose 2}$. This method works equally for any field.