It's an interesting question, naturally there's no correct answer.
I can't remeber exactly how the museum works, but I think it would be sufficient to store all the data in one big DisplayCase enum (position, config val, questions etc). Perhaps also create classes encapsulating questions and answers, and other intricacies. I'd do it this way because 1: all display cases are the same, thus there's really no need for inheritance, and 2: All the data is in one place so it's easier to manipulate when needed.
You can then, as you suggested, switch through the current config value to determine the corresponding display case, and act on the data respectively.
Good luck!
Apa