Let's say you have 5 boxes (representing the questions)
How many different things can you put in the first box? (Representing the answers)
Next box... 4.
And so on.
4×4×4×4×4=4^5=1024 possible ways.
I usually use this reasoning for every combinatorics problem, since I don't get confused with permutation - combination ideas and such.
Here, the order matters, since putting an A in the first box is not the same as putting an A in the second box.
I can see from your reasoning that you took a strange idea, like picking the answer and then the questions, this is wrong since you only pick one answer, you don't really pick the question, the question only marks the possibilities of the answers.
Sorry if something is unclear, I am not an expert at combinatorics