Arrangements of a standard deck of $52$ playing cards if the denominations of the cards are ignored, so that only the suits are distinguished?

Question

I am struggling to see why my solution is wrong for this question.

My solution:

Number of arrangements of length $k$ is $4^k$ and,

since the maximum length is $52$ we have that the total possible ways is $4^1 + \ldots + 4^{52}$.

Answer 1

There are two problems with your answer. The first is you ignore the fact that there are exactly $13$ of each suit. The number of arrangements of $k$ cards is $4^k$ for $k \le 13$, but for $k=14$ it counts arrangements with $14$ of the same suit, so the correct answer is $4^k-4$. The discrepancy grows as $k$ gets larger. If you look for permutations of a multiset you can find information.

The second is that the question requires you have all $52$ cards in the arrangement, so you should not sum over $k$.