Suppose there are $n$ points. Consider the following mapping that takes circles in the plane to points in $\mathbb{RP}^3$. First, use a stereographic projection to map the circles to circles on a sphere. Define the image of the circle to be its center's inversion about the sphere. This mapping satisfies the following important properties: Generalized circles get continuously bijected to points strictly outside the sphere. The set of all circles that contain a point on their exterior get mapped to a plane. Moreover, a circle contains the point on its interior iff its image lies on one side of this plane. After bijecting the $n$ points to $n$ planes, it suffices to determine the number of regions into which $n$ planes divide space. This quantity is A000125 and is equal to $\tbinom{n}{0}+\tbinom{n}{1}+\tbinom{n}{2}+\tbinom{n}{3}.$ An inductive argument to show this value is given at the link.