The Mathematics of Shuffled Cards

It is said that each time you shuffle a 52-card deck,  each arrangement you make may have never existed in all history, or may never exist again. Why? Because of the enormous number of arrangements that can be made using 52 distinct objects (in this case, cards).

To understand this, we can look at the number of arrangements that can be made with smaller number of objects. Lets start with 3 objects A, B, and C. The possible arrangements are ABC, ACB, BAC, BCA, CAB and CBA. Notice that for the first position, there are 3 possible choices (see figure below). Then, after you made the first choice, there are only 2 possible choices left. And after the second choice, you only have 1 possible choice. This means that the number of arrangements of 3 objects is 3 \times 2 \times 1 = 6

Using the method above, we can calculate the number of arrangements of objects by multiplying the number of objects by all the numbers less than it down to 1. For example, the number of arrangements of 4 objects is 4 \times 3 \times 2 \times 1 = 24 (read Introduction to Permutations for details). Therefore, the number possible arrangements for a 52-deck card is

52 \times 51 \times 50 \times \cdots \times 3 \times 2 \times 1

or the product of 52 and all the integers less than it all the way down to 1. Note that the \cdots symbol is used to denote that there are numbers that are not shown in the multiplication above. If we use a calculator to compute the expression above, we get


or a number starting with 8 plus 67 more digits.

Now, let’s compute how many possible arrangements of cards are made in a year. There are 86400 seconds in a day, so if we assume that on the average there are one thousand 52-deck cards shuffled every second in the world, then, the number of arrangements in a year is 1000 decks/second x 86400 second/day x 365 days =  31 536 000 000.

If we divide 8.0658175e+67 by 31 536 000 000, we can exhaust all the arrangement in approximately 2.55 \times 10^{57} years, about

185 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000

times the age of the universe.

Scientists believe that the universe is only 13.8 billion years  or 1.38 \times 10^{10} years.

So, you see, it is possible that all the arrangements of well shuffled cards may not all exist until the end of the universe.

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