Written by Oscar Lee

Introduction

            How many different ways can you shuffle a deck of cards? Well, for the first card, there’s 52 choices. For the second, 51. For the third, 50, and so on. By the Multiplication Principle, which states that if an event can occur in “m” ways and a second independent event can occur in “n” ways, then the number of ways both events can occur is m*n ways, we find the number of ways to order the cards is 52*51*50*49*…*1. In other words, 52! (factorial). This number comes out to 8065817517094387857166063685640376697528950544088327782400000000000. But this is just a really long number on a page. How big is 52!? How much is 52! Seconds? How can we represent this number in our real world, and how can this help us understand numbers better?

Scott Czpiel’s Scenario

             Let’s think about 52! Seconds. Compared to the puny 31536000 seconds in a year, 52! is astronomically larger. So how can we visualize it? Well, Scott Czepiel wrote an essay to help us visualize this. He says to imagine the following scenario: “Start a timer that will count down the number of seconds from 52! to 0.” Then, he says that we will walk around the Earth’s equator, but with a twist: you are only allowed to take one step every billion years. Then, after you make your way around the earth (by taking 1 step every billion of years), you take one drop of water out of the Pacific Ocean. Then, you repeat the process of walking around the equator, and everytime you walk around, you keep draining one singular drop of water. After the ocean is fully drained, you refill the ocean and put a piece of paper underneath you. Now, you once again repeat this process of walking, draining, and placing papers. After your stack of papers has reached the Sun, you repeat another 1000 times. After all this, you have completed just about a third of the timer. 

Figure 1

Micheal Stevens (Vsauce)’s Visual Representation of Czpiel’s Scenario

Source: Vsauce Youtube Channel

52! Atoms

            Okay, we’ve started to get a grasp of how big 52! is with our “ridiculous” scenario. But what about trying to consider something on this earth? Well, 52! is 8.0658175 x 10^67. To try and put this into our real-world context, let’s consider some of the smallest things we can get our hands on: atoms. By compiling the number of every type of element on this earth, and the amount of atoms, we are able to come up with an approximation for the amount of atoms in the world, about 10^50 atoms on Earth. This is still 10^17 times smaller than 52!. So if you gathered up the atoms of every object, animal, and even our own atmosphere, it still wouldn’t come close to 52!.

Figure 2

Number of atom

s in the world written out

Source: sciencenotes.org 

Conclusion

             Whether it be walking around the earth, filling the Pacific Ocean, or stacking paper to the sun, I think we’ve established that 52! is an obscenely large number. Recall that the reason we chose to study this number is because it is the number of permutations or shuffles of a standard deck of cards. This means that every time that YOU shuffle a deck of cards, you are almost certainly discovering a new shuffle that has never been seen before in the history of this universe, which is pretty neat. By understanding the true size of a number like 52!, we can understand the solutions we get in our math competitions, problem sets, take homes, and more generally the nature of large numbers in our natural world. Being able to apply mathematics to the real world is the end goal of mathematics, after all. Trying to wrap our heads around unfathomably large numbers can seem crazy, but it’s this craziness that makes the infinite nature of mathematics so captivating. 

References and Sources

Frauenfelder, M. (2017, March 2). How to imagine 52 factorial. Boing Boing. https://boingboing.net/2017/03/02/how-to-imagine-52-factorial.html

Vsauce. (2016). Math Magic [YouTube Video]. In YouTube. https://www.youtube.com/watch?v=ObiqJzfyACM

Lee, C. (2018, June 28). There are more ways to arrange a deck of cards than there are atoms on Earth. Office for Science and Society.                    https://www.mcgill.ca/oss/article/did-you-know-infographics/there-are-more-ways-arrange-deck-cards-there-are-atoms-earth

 Helmenstine, A. (2022, May 10). How Many Atoms Are in the World? Science Notes and Projects. https://sciencenotes.org/how-many-atoms-are-in-the-world

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