More orders than atoms nearby
Has Anyone Ever Had This Hand Before?
The number of ways to arrange a pack is so large that every good shuffle is almost certainly new.
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Step 1 of 5
Every card taken reduces the choices.
Fifty-two cards could go first. Whichever went, fifty-one could go second. The choices shrink but they multiply together.
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Step 2 of 5
Multiplying is what makes it enormous.
Adding fifty-two numbers gives a few thousand. Multiplying them gives a number with sixty-eight digits, and that difference is the whole point.
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Step 3 of 5
The number is beyond any comparison.
It is vastly more than the atoms in the Earth. Every human who has ever shuffled has not made a dent in the possibilities.
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Step 4 of 5
So a proper shuffle is almost certainly unique.
Every shuffle ever dealt, by everyone, is a rounding error against that number. Not guaranteed — just so overwhelmingly likely that no shuffle in history has ever needed to repeat one.
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Step 5 of 5
Which is why shuffling badly matters.
A new pack in order needs about seven riffle shuffles to genuinely scramble. Three leaves enough of the old order to be readable.
Casinos use machines partly because human shuffling is far less thorough than it looks.
The short version
A pack has a 68-digit number of possible orders, so a properly shuffled deck has almost certainly never existed before.
Try it yourself
Work out the orders of just five cards: 5 × 4 × 3 × 2 × 1 = 120. Now imagine doing that fifty-two times.