Lastly, let's take a look at some rather big numbers: How many ways are there to shuffle a deck of 52 cards
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For the first card you have 52 options, for the second 51, for the third 50 and so on. Thus the total number of different ways a deck can be shuffled is 52 × 51 × 50 × … × 1 – or 52! (factorial).
This number is mindbogglingly huge
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It has 68 digits and if you like tongue twisters, please try to pronounce it: 80 unvigintillion 658 vigintillion 175 novemdecillion 170 octodecillion 943 septendecillion 900 sexdecillion.
Here's the full number:
80,658,175,170,943,900,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
The number of ways to shuffle a single deck of cards is so huge that whenever you shuffle a deck you are virtually guaranteed to have a shuffle that has never been played before and never will be played again.
Number of Possible Different Poker Games
Interestingly enough, the number of different card distributions for poker games is much smaller.
The bottom cards of the deck are not used and thus it doesn't matter how they are shuffled. As a matter of fact, for a heads-up game of Hold‘em, you only use 9 cards – 4 hole cards and 5 for the board
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The total number of different distributions for a heads-up Hold'em game is a bit over 1 trillion: 1,390,690,501,200 – a number much smaller than 52!, but still large enough that you'll never see the same heads-up game twice in your lifetime.
