Father and son play a game with a 52-card deck of cards. Each card of this deck has a number from 1 to 13 on it and there are 4 cards exactly the same for each number.
In how many different ways can these cards be placed in the deck?
My effort: I first began by considering the simpler problem where there are only 4 exact same aces, other than that the deck would be standard (2 colors, 4 symbols, 52 cards) except for the aces. To find all the possible different ways this specific deck can be arranged, I thought of considering the 4 aces as distinct ones, and then dividing by 4! since that's the number of different ways 4 items can be placed without repetitions.
Having solved the relaxed problem, I thought that by induction that the answer to the original question would be $\frac{52!}{(4!)^{13}}$.
Is my answer correct? Am I missing something big? Any help would be greatly appreciated.