Final answer:
The most important factor determining whether a collapsing dense core in an interstellar cloud becomes a single-star or multiple-star system is the amount of rotation (spin) of the region.
Step-by-step explanation:
The question pertains to the factors that determine whether a collapsing region within an interstellar cloud will become a single-star or a multiple-star system. During the collapse of a dense core into a protostar, as the gravitational force strengthens and the core's density increases, a key factor in the outcome is the amount of rotation (spin) of the collapsing region. High rotation can lead to fragmentation or the formation of binary or multiple star systems due to conservation of angular momentum, which results in surrounding material flattening into a disk. This disk can fragment further, leading to the formation of multiple stellar objects gravitationally bound together.