Fletcher Hays

2022-06-24

Arrangements of a standard deck of 52 playing cards if the denominations of the cards are ignored, so that only the suits are distinguished?
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 ${4}^{1}+\dots +{4}^{52}$.

Belen Bentley

Expert

Step 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.
Step 2
The second is that the question requires you have all 52 cards in the arrangement, so you should not sum over k.

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