Why a well-shuffled deck is almost certainly a first

The number of ways to order 52 cards is so large that the total number of shuffles ever performed is a tiny fraction of it. If a shuffle is truly random, the chance that its result matches any earlier shuffle in history is vanishingly small. The claim holds for well-shuffled decks, not for every shuffle. [1] [2]

Any of 52 cards can go first, then 51 remain for the second position, then 50, and so on. Multiplying these choices gives 52 factorial, written 52!. As [math.hmc.edu](https://math.hmc.edu/funfacts/making-history-by-card-shuffling/) notes, this is larger than 8 x 10^67. [mathsy.io](https://www.mathsy.io/articles/deck-of-cards-shuffles/) gives the more precise figure of about 8.07 x 10^67, a number with 68 digits. [1] [2]

[Scientific American](https://www.scientificamerican.com/article/how-the-math-of-shuffling-cards-almost-brought-down-an-online-poker-empire/) puts the scale in physical terms: it is more than a quadrillion times the number of atoms on Earth. [3]

The HMC fun fact offers a time comparison: someone shuffling once per second since the universe began, believed to be about 14 billion years ago, would not have made more than 10^18 shuffles. That is a minuscule share of 52!. [1]

Scientific American stresses that 1/52! only gives the odds of hitting one specific order. The real question is whether any two shuffles ever match, a collision problem like the birthday paradox. Its calculation for eight billion people each shuffling once gives a chance of a match that differs from zero only around the 47th decimal place of a percent. With each person shuffling 100 decks, it is still below 8 x 10^-43 percent. It also considers all people who have ever lived, estimated at about 117 billion, and reaches the same conclusion. [3]

Mathsy.io makes a deliberately generous estimate of 10^20 shuffles in history and gets a repeat probability of roughly 6 x 10^-29. [2]

All of this assumes a genuinely random shuffle. Not every shuffle involves randomness, and a single random riffle does not make every order equally likely. HMC says one riffle can still produce nearly 2^52 configurations, about 4.5 x 10^15, and that after seven random riffles nearly every configuration is about equally likely. Mathsy.io cites Bayer and Diaconis for the seven-riffle result and warns that new decks all start in the same factory order, so two lazy single riffles could plausibly match. [1] [2] [4]