How many ways can the 52 cards of a standard deck be arranged into a circular permutation if all of the suits must be kept together in their own groups?
This is a question I gave on a homework on a combinatorics lesson I'm teaching, but I just realized I'm unsure on the best answer. I'm pretty competent in general but this is the first time I've taught the course.
The sources I have been using suggest that 3!13!13!13!13! is the answer (3! ways to put 4 groups into a circular arrangement, then the 13! for the permutations of the cards in each suit) but I am second guessing myself.
If I instead wonder how many ways I can put the cards into a linear permutation with the same "suits all together" restriction, I am absolutely certain that there are 4!13!13!13!13! distinct arrangements. These strings would all be 52 cards long. Moving the first card to the last position gives a new linear permutation, but both of these lines can be bent into the same circle. I fact, I can repeat this process and still get another instance of this-line-makes-the-same-circle, so on and so on until I loop back to the line I started with. This means that counting the linear permutations overcounts the circular permutations by a factor of 52, and the best answer is actually 4!13!13!13!13!/52.
So which is it?
TLDR is the answer 3!13!13!13!13! , 4!13!13!13!13!/52, or something else entirely?