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You’ll Never Look at a Deck of Cards the Same Way Again

By Parth Mehta

It all starts with a simple deck of playing cards. These 52 thin cardboards with colorful designs printed on their sides seem pretty much harmless enough. Yet as they say, the complexity of things begins from the simplest systems. So, A harmless question. In how many ways can they be arranged?

For this we need to understand the mathematical operation factorial (!). In simple terms, Factorial is the operation of multiplying any natural number with all the natural numbers that are smaller than it, For Example: 4! = 4 × 3 × 2 × 1 = 24

You can visualize the arrangements by constructing a randomly generated shuffle of the deck. Start with all the cards in one pile. Randomly select one of the 52 cards to be in position 1. One of the remaining 51 cards for position 2, then one of the remaining 50 for 3, and so on. Hence, the total number of ways you could arrange the cards is 52 × 51 × 50 ×… ×3 ×2 ×1, or 52! I think the exclamation mark, the symbol for the factorial operator is to highlight the fact that this function produces surprisingly large numbers in a very short time! Factorial intended 🙂

The richness of most card games owes itself to this fact that the number of variations that these 52 cards can produce is virtually endless. If you have a pocket calculator, that maxes out at 12digits, an attempt to calculate the factorial of any number greater than 14 results only in value of “Error”. So, if 15! will break a typical calculator, how large is 52!? Who says you can’t create history? It’s as simple as shuffling cards.

Would you believe every time you gave the deck a proper shuffle, you were holding a sequence of cards which never existed in the history of all mankind? No one has, and likely never, ever held the exact same arrangement of 52 cards you just did.

Shall we play a game? Let’s Permute this!

  • Start a timer that will count down the number of seconds from 52! to 0.
  • Choose your favorite spot on the earth’s equator. You have to circumvent the Earth along the equator. But there’s a catch — you can only take one step every billion years. [A billion years equals 3.155 × 1016seconds]

(Did I lock the door before leaving?)

  • After you complete one journey around the equator and reach the point where you started, remove one drop of water from the Pacific Ocean. 

[The equatorial circumference of the Earth is 40,075,017 meters, according to WGS84]

  • Now repeat the whole process — walk around the Earth at one billion years per step, removing one drop of water from the Pacific Ocean every time you circle the globe. Continue until the Pacific Ocean is empty. 

[There are 20 drops of water per milliliter, and the Pacific Ocean contains 707.6 million cubic kiloliters of water]

(Is there a full stop to this?)

  • After you have emptied the Pacific Ocean, Pick a sheet of paper and place it flat on the ground. Fill the ocean back up and start the entire process all over again, adding a sheet of paper to the stack every time the ocean is emptied. Do this until the stack of paper reaches the Sun. Once the stack reaches the Sun take it down and do it all over again. One thousand times more.

(How. Long. Has. This. Been. Going. On!?)

Surely the timer must have reached 0 by now?


You’re only done with one-third of the time!

(And you thought afternoons were boring)

  • To pass the remaining time, start shuffling your deck of cards. Every billion years deal yourself a 5-card poker hand. Each time you get a royal flush, buy yourself a lottery ticket. A royal flush occurs in one out of every 649,740 hands.
  • If that ticket wins the jackpot, throw a grain of sand into the Grand Canyon. Keep going and when you’ve filled up the canyon with sand, remove one ounce of rock from Mt. Everest. Now empty the canyon and start all over again. 
  • When you’ve levelled Mt. Everest, look at the timer. You still have 5.364 × 1067seconds remaining. Mt. Everest weighs about 357 trillion pounds. You barely made a dent. If you were to repeat this 255 times, you would still be looking at 3.024 × 1064seconds. The timer would finally reach zero sometime during your 256th attempt. Exercise for the reader: at what point exactly would the timer reach zero?

Back to the seat comrades!

Sorry to break it to you, Of course, in reality none of this could ever happen. The truth is, the Pacific Ocean will boil off as the Sun becomes a red giant before you could even take your fifth step in your first trek around the world. However, Somewhat more of an obstacle is the fact that all the stars in the universe will eventually burn out leaving space a dark, ever-expanding void by the time you remove a milliliter of water from the pacific (Not to forget the pacific is already boiled-off).


By the way,

52! = 52 × 51 × 50 ×… ×3 ×2 ×1= 80658175170943878571660636856403766975289505440883277824000000000000

[Visualization and statistics from Scott Zcepiel]


Published by mscnm

Add some excitement, Minus the boredom, multiply your skills, divide the stress. Math is an integral part of our lives, so connect with MSCNM where the fun is derived

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