It's unlikely binary or multiple systems will arise as the result of a dense collisional environment such as in the center of a star cluster. Bionaries form there, but for totally different reasons. Two single stars will either approach and then slingshot off into space, or they will collide, they will not form a binary, because of momentum transfer issues. You can have a near simultaneous collision of three bodies to form a binary, the third menber being slung off with the excess momentum. It is dynamically possible to form triple and N-member systems, but extremely difficult to do so in practice.

Binaries are the result of hierarchical collapse of interstellar clouds. Only two-body systems are stable, n-body systems are again, not impossible, but chaotically unstable. So how do we get n-body systems in the real world? This is because a two-body system can easily have a two-body system as one of it's members. These are based on the rule of twos: Two stars orbit a common center of gravity. Triple systems form when the third member orbits the COG of the other two. A quadruple system is either two binaries orbiting each other, or two separate stars orbiting the COG of the other two. And so on. The last time I looked, the record holder of all multiple systems contained eleven members! Members with 3, 4, 5 or 6 are quite common.

This process occurs when the original cloud collapses, binaries easily form, but there is no reason a third star can't orbit a binary, or two binaries orbit each other, and so on. In the event the primary is much more massive than all the secondaries, you get "solar systems" or multiple mooned systems like our own gas giants. Sytems like this can be broken up in a dense cluster environment by near-collisions, but encounters don't cause them. This idea of hierarchical collapse seems to be the result of a strange wrinkle in Newtonian Mechanics, N-body systems only form under special conditions: multiples of twos.