Is "Cary is a drummer in a band" a run off sentence or fragment?

It is a fragment sentence.

One coluld argue that this is actually a complete sentence. It has both a subject and predicate, and makes sense on it's own as an independant clause, but seeing as that's not one of your options, I suppose you should choose a fragment sentence: A phrase of incomplete thought. I can't be a run - on sentence, because two independant clauses aren't combined.