Is every shuffled deck unique? Almost, after 7 riffles

A deck of 52 cards shuffled thoroughly, which means about seven riffle shuffles, is almost certainly in an order that has never existed before. There are 52! (52 factorial) possible orders, roughly 8 × 10⁶⁷, a number with 68 digits, and no amount of human card-playing comes close to exhausting them.

How many ways can a deck of cards be ordered?

Exactly 52!, and the count is simple to build. The first card can be any of 52, the second any of the remaining 51, the third any of 50, and so on down to 1. Multiply the choices together and the total dwarfs the estimated number of atoms making up the Earth.

To feel the gap: Harvey Mudd College points out that someone shuffling once a second since the Universe began, about 14 billion years ago, would have managed fewer than 10¹⁸ shuffles. Multiply that by billions of card players and the total is still a vanishingly small fraction of 52!, so a genuinely random order is overwhelmingly likely to be new.

How many shuffles does it take to randomise a deck?

About seven riffles, and it matters, because only a truly random order is likely to be new. A new deck comes out of the box in a fixed order, and a few casual shuffles leave it far from random, so lightly shuffled decks are nothing like unique. In 1992, mathematicians Dave Bayer and Persi Diaconis showed that after seven riffle shuffles every arrangement of a 52-card deck is nearly equally likely. With fewer, the deck stays far from random; with more, it barely improves. The overhand shuffle is far worse: it takes about 2,500 of them.

Their proof assumed a disciplined shuffle, with the deck cut roughly in half each time. In June 2026, Quanta Magazine reported that Harvard statistician Mark Sellke and graduate students Jialu Shi and Jiamin Wang had proved that this sudden tip into randomness, called a cutoff, still happens with sloppier riffles where the deck is not cut evenly. “It’s a brilliant piece of mathematics,” Diaconis said.