Getting the name of combinatorial problems

Solution 1:

One way is to calculate the terms for small $n$ (in your example the size of the smallest set with your desired property) and then look these up in the On-Line Encyclopedia of Integer Sequences

Solution 2:

There is the twelve-fold way, which is a classification of several of the most common combinatorial problems along the lines you are asking about in your edit. If you have $n$ balls to place into $x$ boxes, you can have the balls labeled or not, the boxes labeled or not, and whether the mapping of balls to boxes has no restrictions (choosing the boxes "with replacement," in some contexts), is injective (no more than one ball per box, or choosing the boxes "without replacement," in some contexts), or is surjective (at least one ball per box). Considering all these possibilities gives the twelve options in the name of the classification.

See also Richard Stanley's Enumerative Combinatorics, Vol. I, where it is discussed in detail.

Added (in response to OP's request for generalization): On p. 88 of Stanley's text (which you can download directly from his site using the link above) he says, "There are many possible generalizations of the Twelvefold Way and its individual entries." He then goes on to discuss one of them. In the notes (p. 107) on the chapter in which the Twelvefold Way appears Stanley also says, "An extension of the Twelvefold Way to a 'Thirtyfold Way' (and suggestion of even more entries) is due to R. Proctor." Tracking down the reference leads to Proctor's article "Let's Expand Rota's Twelvefold Way For Counting Partitions!", which I'm not familiar with but which certainly appears to be the kind of thing you're asking for.