It looks like probably there is some mistake in your description of the problem. If you want to show that {A,B,C,D} is enumerable you just need to show an injective map from this set to the natural numbers (or something like that, depending on what your lecture notes say). This is easy and isn't made any easier by considering the strings over the alphabet. You just map A to 1, B to 2, C to 3, D to 4.
Probably you mean to say something like: Prove that the power set of {A,B,C,D} is enumerable. The powerset can be identified with the set of all strings that are in alphabetical order, such that each string contains at most one occurrence of each element in the alphabet. If perhaps your lecture notes tell you that the set of all strings over an enumerable alphabet is always enumerable, then we can identify the powerset with a subset of those strings. If we further know (say, somewhere in your lecture notes) that a subset of an enumerable set is enumerable, then this would be all we need to draw the desired conclusion.
But of course there's a lot of guessing here, since I don't know what's in your lecture notes.
