If you shuffle a deck of cards well, chances are the deck is in a combination that no other deck of cards has ever been in before, and no other deck will ever be in again.
This blows my mind, and thought I'd share.
https://czep.net/weblog/52cards.html
The number of possible permutations of 52 cards is 52!. I think the exclamation mark was chosen as the symbol for the factorial operator to highlight the fact that this function produces surprisingly large numbers in a very short time. If you have an old school pocket calculator, the kind that maxes out at 99,999,999, an attempt to calculate the factorial of any number greater than 11 results only in the none too helpful value of "Error". So if 12! will break a typical calculator, how large is 52!?
52! is the number of different ways you can arrange a single deck of cards. You can visualize this 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. Next, randomly select one of the remaining 51 cards for position 2, then one of the remaining 50 for position 3, and so on. Hence, the total number of ways you could arrange the cards is 52 * 51 * 50 * ... * 3 * 2 * 1, or 52!. Here's what that looks like:
80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000
This number is beyond astronomically large. I say beyond astronomically large because most numbers that we already consider to be astronomically large are mere infinitesimal fractions of this number. So, just how large is it? Let's try to wrap our puny human brains around the magnitude of this number with a fun little theoretical exercise. Start a timer that will count down the number of seconds from 52! to 0. We're going to see how much fun we can have before the timer counts down all the way.
Shall we play a game?
Start by picking your favorite spot on the equator. You're going to walk around the world along the equator, but take a very leisurely pace of one step every billion years. Make sure to pack a deck of playing cards, so you can get in a few trillion hands of solitaire between steps. After you complete your round the world trip, remove one drop of water from the Pacific Ocean. Now do the same thing again: walk around the world at one billion years per step, removing one drop of water from the Pacific Ocean each time you circle the globe. Continue until the ocean is empty. When it is, take one sheet of paper and place it flat on the ground. Now, fill the ocean back up and start the entire process all over again, adding a sheet of paper to the stack each time you’ve emptied the ocean.
Do this until the stack of paper reaches from the Earth to the Sun. Take a glance at the timer, you will see that the three left-most digits haven’t even changed. You still have 8.063e67 more seconds to go. So, take the stack of papers down and do it all over again. One thousand times more. Unfortunately, that still won’t do it. There are still more than 5.385e67 seconds remaining. You’re just about a third of the way done.
And you thought Sunday 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. 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 5.364e67 seconds remaining. You barely made a dent. If you were to repeat this 255 times, you would still be looking at 3.024e64 seconds. The timer would finally reach zero sometime during your 256th attempt.
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Of course, 52! is nowhere close to being as large as Graham's number, and there are even larger numbers with specific names/uses, but it's still a very cool way to explain how large 52! is.
This blows my mind, and thought I'd share.
https://czep.net/weblog/52cards.html
The number of possible permutations of 52 cards is 52!. I think the exclamation mark was chosen as the symbol for the factorial operator to highlight the fact that this function produces surprisingly large numbers in a very short time. If you have an old school pocket calculator, the kind that maxes out at 99,999,999, an attempt to calculate the factorial of any number greater than 11 results only in the none too helpful value of "Error". So if 12! will break a typical calculator, how large is 52!?
52! is the number of different ways you can arrange a single deck of cards. You can visualize this 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. Next, randomly select one of the remaining 51 cards for position 2, then one of the remaining 50 for position 3, and so on. Hence, the total number of ways you could arrange the cards is 52 * 51 * 50 * ... * 3 * 2 * 1, or 52!. Here's what that looks like:
80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000
This number is beyond astronomically large. I say beyond astronomically large because most numbers that we already consider to be astronomically large are mere infinitesimal fractions of this number. So, just how large is it? Let's try to wrap our puny human brains around the magnitude of this number with a fun little theoretical exercise. Start a timer that will count down the number of seconds from 52! to 0. We're going to see how much fun we can have before the timer counts down all the way.
Shall we play a game?
Start by picking your favorite spot on the equator. You're going to walk around the world along the equator, but take a very leisurely pace of one step every billion years. Make sure to pack a deck of playing cards, so you can get in a few trillion hands of solitaire between steps. After you complete your round the world trip, remove one drop of water from the Pacific Ocean. Now do the same thing again: walk around the world at one billion years per step, removing one drop of water from the Pacific Ocean each time you circle the globe. Continue until the ocean is empty. When it is, take one sheet of paper and place it flat on the ground. Now, fill the ocean back up and start the entire process all over again, adding a sheet of paper to the stack each time you’ve emptied the ocean.
Do this until the stack of paper reaches from the Earth to the Sun. Take a glance at the timer, you will see that the three left-most digits haven’t even changed. You still have 8.063e67 more seconds to go. So, take the stack of papers down and do it all over again. One thousand times more. Unfortunately, that still won’t do it. There are still more than 5.385e67 seconds remaining. You’re just about a third of the way done.
And you thought Sunday 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. 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 5.364e67 seconds remaining. You barely made a dent. If you were to repeat this 255 times, you would still be looking at 3.024e64 seconds. The timer would finally reach zero sometime during your 256th attempt.
***********************************************************************
Of course, 52! is nowhere close to being as large as Graham's number, and there are even larger numbers with specific names/uses, but it's still a very cool way to explain how large 52! is.
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