--- title: "How many ways can you shuffle a deck of cards? about 8 × 10^67" description: "about 8 × 10^67 How the figure is estimated, how confident it is, and what makes the scale graspable." url: https://atoneplace.net/how-many/possible-shuffles-of-a-deck/ site: "At One Place" --- # How many ways can you shuffle a deck of cards? **How many ways can you shuffle a deck of cards?: about 8 × 10^67** If every one of the 8 billion people alive dealt out a new deck order every second, and had been doing so since the Big Bang, they would between them have produced about 3.5 × 10^27 orderings — roughly one forty-thousandth of one trillionth of one trillionth of one percent of the total. A 52-card deck can be arranged in 52! ways — 52 × 51 × 50 × … × 1 — which is exactly 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000. The practical consequence is the interesting bit. Every time you properly shuffle a deck, you almost certainly produce an ordering that has never existed before in the history of the world and never will again. Not "probably" in a loose sense — the probability that any given shuffle repeats a previous one is so close to zero that it is not worth writing down. ## How the figure is arrived at 52 choices for the first card, 51 for the second, and so on: the factorial. Worth knowing alongside it: seven riffle shuffles are enough to randomise a deck, and six are not. That result, from a 1992 paper by Bayer and Diaconis, shows a sharp cutoff — the deck goes from clearly ordered to effectively random within a single shuffle. ## Other numbers and possibilities counts - [How many combinations does a Rubik's Cube have?](https://atoneplace.net/how-many/rubiks-cube-states/) — 43,252,003,274,489,856,000 — about 43 quintillion - [How many possible chess games are there?](https://atoneplace.net/how-many/possible-chess-games/) — about 10^120 games; about 4.8 × 10^44 legal positions - [How many combinations are there in the lottery?](https://atoneplace.net/how-many/lottery-combinations/) — 292,201,338 for Powerball - [How many digits of pi have been calculated?](https://atoneplace.net/how-many/digits-of-pi-computed/) — about 202 trillion - [How many possible passwords are there?](https://atoneplace.net/how-many/possible-passwords/) — 6.6 quadrillion for eight characters — 540 sextillion for twelve - [How many Sudoku puzzles are there?](https://atoneplace.net/how-many/possible-sudoku-grids/) — 6,670,903,752,021,072,936,960 completed grids - [How many prime numbers are there under a million?](https://atoneplace.net/how-many/primes-under-a-million/) — exactly 78,498 - [How many possible positions are there in Go?](https://atoneplace.net/how-many/possible-go-positions/) — about 2.08 × 10^170 ## Common questions ### How many ways can you shuffle a deck of cards? about 8 × 10^67 ### How is that estimated? 52 choices for the first card, 51 for the second, and so on: the factorial. Worth knowing alongside it: seven riffle shuffles are enough to randomise a deck, and six are not. That result, from a 1992 paper by Bayer and Diaconis, shows a sharp cutoff — the deck goes from clearly ordered to effectively random within a single shuffle. ## Related pages - [How many combinations does a Rubik's Cube have?](https://atoneplace.net/how-many/rubiks-cube-states/) - [How many possible chess games are there?](https://atoneplace.net/how-many/possible-chess-games/) - [How many combinations are there in the lottery?](https://atoneplace.net/how-many/lottery-combinations/) - [How many digits of pi have been calculated?](https://atoneplace.net/how-many/digits-of-pi-computed/) - [How many possible passwords are there?](https://atoneplace.net/how-many/possible-passwords/) - [How many Sudoku puzzles are there?](https://atoneplace.net/how-many/possible-sudoku-grids/) - [How many prime numbers are there under a million?](https://atoneplace.net/how-many/primes-under-a-million/) - [How many possible positions are there in Go?](https://atoneplace.net/how-many/possible-go-positions/) - [How many people are there in the world?](https://atoneplace.net/how-many/people-alive-right-now/) - [How many stars are there in the universe?](https://atoneplace.net/how-many/stars-in-the-universe/) - [How many countries are there in the world?](https://atoneplace.net/how-many/countries/) - [How many bones are in the human body?](https://atoneplace.net/how-many/bones-in-the-human-body/) - [How many grains of sand are there on Earth?](https://atoneplace.net/how-many/grains-of-sand-on-earth/) - [How many ants are there in the world?](https://atoneplace.net/how-many/ants-on-earth/) ## Sources - Calculated on this page — At One Place (/about/)