Here, the factorials you mention don't have anything to do with probability - rather, they have to do with combinatorics. This is because in this context, you are talking about uniformly random elements of some set - and so counting up the elements with a specific property is a necessary step of the process.

Factorials are important because $n!$ is the number of ways to list - in order - a set of $n$ objects that are distinguishable. Because of this, it also comes up in other arrangements - such as the number of ways to choose $k$ elements from a set of $n$ (in an order or otherwise). Further, many other combinatorial constructions can be carried out starting with these basic ideas of arrangements, and so they involve factorials as well.

You are absolutely right that distinguishable is a keyword that screams factorial; however, distinguishability of the objects is often implicit in the problem, rather than explicitly stated. The big thing: look for any situation in which you are arranging objects in some way.