If I place place $N$ particles in a sphere of radius $R$, selecting positions across the sphere's volume with uniform probability, what is the exact probability distribution for the number of pairwise distances between points less than or equal to some distance $d$? Is an exact distribution known for rectangular or cylindrical boundary conditions?

Please note that I am looking for an exact probability distribution provided spherical boundary conditions. One can of course come up with a "good enough" approximation for some physical system, perhaps with the aid of simulation. But I'd like to understand how, if possible, one can arrived at a closed form expression for the distribution.

Update: To ask the question in a different way, what's the probability distribution for the number of particles that have a lowerbound nearest-neighbor distance $\geq d$?