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