The spatial analogue of the Apollonian tangency problem: to construct spheres that are tangent to 4 given spheres. If we accept the special cases of a sphere with zero radius, i.e., a point, and a sphere with infinite radius, i.e., a plane, then how many such - in this generalized sense - spatial Apollonian problems are there?
In every spatial Apollonian problem, 4 given elements appear, which are selected from 3 elements (point, plane, sphere). (For example: a sphere is to be constructed passing through a given point and touching two given spheres and a given plane.) Since the order is irrelevant when selecting these elements and there can be repetitions among the selected elements, the number of all possible problems is given by the number of 4th class repetitive combinations of 3 elements, that is,

$$
C_{3}^{i, 4}=\frac{3 \cdot 4 \cdot 5 \cdot 6}{1 \cdot 2 \cdot 3 \cdot 4}=15
$$

Thus, there are 15 types of generalized spatial Apollonian problems.