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

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

Sources

  1. Calculated on this page — At One Place

How these figures are compiled and checked