Number of ways to separate $n$ points in the plane
Let there be $k$ oriented lines that divide the $n$ points into different left and right sides.
There are $2$ trivial lines and $k-2$ nontrivial lines. Let each nontrivial line swing counter-clockwise as far as it can, until it hits two of the points. The nontrivial lines will match with the ordered pairs of the points, of which there are $n(n-1)$. So $k=n^2-n+2$.
Or, if you want to disregard orientation, divide by two! $\frac{k}{2}=\binom{n}{2}+1$.