Shuffling Cards

Every time you shuffle a deck of playing cards, it’s likely that you have come up with an ordering of cards that is unique in human history. For example, I shuffled a deck of cards this afternoon, and my friend Adam split the deck, and this is the order that the cards came out it.

How many different orders are there?

There are 52 cards in a deck of cards. Imagine an “ordering of cards” as 52 empty spots to be filled:

How many different possibilities are there for what could go in the first spot? The answer is 52 – any of the 52 cards could go there. What about the second spot? Now that you’ve already chosen a card for the first spot, there are only 51 cards left, so there are only 51 different possibilities for the second spot. And for the third spot, we only have 50 choices.

If we stop there, and just fill up the first three spots, that’s like asking how many different possibilites there are for dealing three cards in order. Here’s one of the possibilites:

How many different possible combinations are there for three cards in order? We just multiply how many possibilities there were for the first position (52) with the possibilities for the second position (51) with the possibilities for the third position (50). So there are 52 • 51 • 50 = 132600 different possibilites for three cards in order.

What about a whole deck? We just multiply the possibilities for each of the 52 positions, which is 52 • 51 • 50 • 49 • 48 • 47 • 46 • 45 • 44 • 43 • 42 • 41 • 40 • 39 • 38 • 37 • 36 • 35 • 34 • 33 • 32 • 31 • 30 • 29 • 28 • 27 • 26 • 25 • 24 • 23 • 22 • 21 • 20 • 19 • 18 • 17 • 16 • 15 • 14 • 13 • 12 • 11 • 10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1. A mathematical way of representing all those numbers multiplied together is called the factorial (See description on MathWorld), so we could write this as 52!, which means the same thing. When you multiply all those numbers together, you get 80658175170943878571660636856403766975289505440883277824000000000000. That number is 68 digits long. We can round off and write it like this: 8.0658X1067.

How many times have cards been shuffled in human history?

That’s an impossible number to know. So let’s overestimate. Currently, there are between 6 and 7 billion people in the world. Also, the modern deck of 52 playing cards has been around since 1300 A.D. probably. If we assume that 7 billion people have been shuffling cards once a second for the past 700 years, that will be way more than the actual number of times cards have been shuffled. 700 years is 255675 days (plus or minus a couple for leap year centuries), which is 22090320000 seconds. Now, if 7000000000 people had been shuffling cards once a second for 22090320000 seconds, they would have come up with 7000000000 • 22090320000 different combinations, or orderings of cards. When you multiply those numbers together you get 154632240000000000000, or rounding off, 1.546X1020.

So, it’s safe to say that in human history, playing cards have been shuffled in less than 1.546X1020 different orders.

Is this order unique in human history?

Probably so. When I shuffled the cards this afternoon, and came up with the order you see in the picture, that is one of 8.0658X1067 different possible orders that cards can be in. However, in the past 700 years since playing cards were invented, cards have been shuffled less than 1.546X1020 times. So the chances that one of those times they got shuffled into the same exact order you see here are less than 1 in 100000000000000000000000000000000000000000000000 (1 in 1047).

At what point do you say something is impossible? If the chances are 1 in 1000? 1 in a million?1 in ten trillion?1 in 1 in 1047? In the movie Dumb and Dumber (See IMDB Info), Lloyd asks Mary what the chances are of the two of them getting together. She replies “1 in a million.” He responds, “so you’re saying there’s a chance?!”

So… if you think there’s a chance that maybe, just maybe somebody, somewhere, at some time may have shuffled a deck of cards just like this ordering you see here, then you’re like Lloyd Christmas in the movie.


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