Suppose there are $n$ unique points on a $2$-sphere with no three points co-great circle. And we can choose a great circle passing through one of the points that partition the rest points into two subsets. If any point passes the great circle is excluded from the final partition.
How many ways can there be for all possible partitions?
For each point there are $n-1$ partitions with $n-1$ points and $n-1$ partitions with $n-2$ points. Summing over all points will count partitions with $n-2$ points twice. So there are $$n(n-1)+n(n-1)/2 = \tfrac32n(n-1)$$ in total.