  Direct collisions between finite-sized particles occur commonly in many areas
of astrophysics. Such collisions are typically mediated by chaotic, bound
gravitational interactions involving small numbers of particles. An important
application is stellar collisions, which occur commonly in dense star clusters,
and their relevance for the formation of various types of stellar exotica. In
this paper, we return to our study of the collision rates and probabilities
during small-number chaotic gravitational interactions ($N$ $\lesssim$ 10),
moving beyond the small-number particle limit and into the realm of larger
particle numbers ($N$ $\gtrsim$ 10$^3$) to test the extent of validity of our
analytic model as a function of the particle properties and the number of
interacting particles. This is done using direct $N$-body simulations of
stellar collisions in dense star clusters, by varying the relative numbers of
particles with different particle masses and radii. We compute the predicted
rate of collisions using the mean free path approximation, adopting the
point-particle limit and using the sticky-star approximation as our collision
criterion. We evaluate its efficacy in the regime where gravitational-focusing
is important by comparing the theoretical rates to numerical simulations. Using
the tools developed in previous papers in this series, in particular Collision
Rate Diagrams, we illustrate that our predicted and simulated rates are in
excellent agreement, typically consistent with each other to within one
standard deviation.
