This has got to be one of my all time favourite mathematical problems
The fact of the matter is, a deck of cards is a very tangible thing; we have all held a deck of cards, played card games, shuffled the deck, and have most likely dealt them all out in some random order.
It takes maybe 30 seconds to lay a shuffled deck of cards out in any random order (give it a try now), since there is only 52 cards how many different arrangements could there possibly be? 100? 1000? maybe 1,000,000?
Not Even Close
There is a 100% chance (almost) that the random order you just dealt has never been dealt before in the history of the world. You have just done something completely unique.
To explain just how many arrangement there are, think for a minute of 3 marbles: 1 green 1 blue and 1 red.
How many ways could you arrange these?
- G, B, R
- G, R, B
- B, G, R
- B, R, G
- R, G, B
- R, B, G
There are six total ways, and this number is called 3 Factorial (3!).
3! = 3 x 2 x 1
Which as we discovered = 6
So if 3 objects can be arranged 3! ways then 52 objects must be able to be arranged 52! ways.
This is where the math starts getting long, because:
52! = 52 x 51 x 50 x 49 x 48 x 47 x 46 x 45 x 44 x 43 x 43 x 41 x 40 x 39 x 38 x 37 x 36 x 35 x 34 x 33 x 32 x 31 x 30 x 29 x 28 x 27 x 26 x 25 x 24 x 23 x 22 x 21 x 20 x 19 x 18 x 17 x 16 x 15 x 14 x 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
This comes to 8.0658 x 10^67
or 80658175170943878571660636856403766975289505440883277824000000000000
Now when written down, while there is a lot of numbers, 52! doesn't seem all that impressive.
To better help put into perspective just how large this number is, a mathematician has outlined exactly what you could get done if you start a timer that counts down the number of seconds from 52! to 0.
What You Can Do in 52! Seconds
Here is a 5 minute video by Michael Mitchell (Vsauce) that explains the same task but with animations, definitely worth a watch!
Thanks for reading