Suppose we have $n$ distinguishable points in $\mathbb{R}^d$. What is $f(n, d)$, the largest number of different ways we can separate them using a single hyperplane? I don't consider swapping the 'left' and 'right' side of the plane to be different.
I found the following question for the $d = 2$ case, so $f(n, 2) = \binom{n}{2} + 1$.
You may assume that the points are in a position allowing the most number of separations. For $d = 2$ it is shown this doesn't matter (beyond no collinearity), but I don't know if this also holds for higher dimensions (with points in general position).
Assuming the $n$ points in general position, as shown in the paper "The number of partitions of a set of N points in k dimensions induced by hyperplanes" by E. F. Harding, the function you are looking for is:
$$f(n,d) = \sum_{k=0}^{d}{n-1 \choose k}$$