How Many Ways Can You Arrange a Deck of Cards?
maths·@benzene·
0.000 HBDHow Many Ways Can You Arrange a Deck of Cards?
<html> <pre><code>This has got to be one of my all time favourite mathematical problems</code></pre> <p> <img src="https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcRBQbBu5TBJhWQJ0F772STSjqPUKPO6cix6lLiD2G6sYk9MrDKHeA" width="225" height="225"/></p> <p>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. </p> <p>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? </p> <p><strong>Not Even Close</strong></p> <p>There is a 100% chance (<em>almost</em>) that the random order you just dealt has never been dealt before in the history of the world. You have just done something <em>completely unique</em>. </p> <p>To explain just how many arrangement there are, think for a minute of 3 marbles: 1 green 1 blue and 1 red.</p> <p><br> <img src="https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcRAu8EdrGHPAt1BAgdHHPh-A074id8zQyVmesBFs8go2F6I0fdJ" width="417" height="121"/></p> <p>How many ways could you arrange these?</p> <ol> <li>G, B, R</li> <li>G, R, B</li> <li>B, G, R</li> <li>B, R, G</li> <li>R, G, B</li> <li>R, B, G</li> </ol> <p>There are six total ways, and this number is called 3 Factorial (3!).<br> 3! = 3 x 2 x 1<br> Which as we discovered = 6<br> <br> So if 3 objects can be arranged 3! ways then 52 objects must be able to be arranged 52! ways.<br> This is where the math starts getting long, because:<br> 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</p> <p>This comes to 8.0658 x 10^67<br> or 80658175170943878571660636856403766975289505440883277824000000000000</p> <p>Now when written down, while there is a lot of numbers, 52! doesn't seem all that impressive.</p> <p>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. </p> <p> <a href="https://czep.net/weblog/52cards.html">What You Can Do in 52! Seconds</a></p> <p>Here is a 5 minute video by Michael Mitchell (Vsauce) that explains the same task but with animations, definitely worth a watch!</p> <p>https://www.youtube.com/watch?v=T69cguFzZ_w</p> <p>Thanks for reading<br> </p> </html>