The order of the cards you've just shuffled ...
... has almost certainly never been seen before
You know that old claim that every time you shuffle a deck of cards, you're probably creating an order that has never been seen before? I decided to hang some numbers on it ...
At the statistic's heart is the staggeringly large number of orders into which 52 cards can be rearranged. This is 52 factorial (written as 52!), or 52 x 51 x 50 x 49 ... all the way down to 1. That's because the first card (in the pictured deck it's the two of clubs) can be any one of 52, leaving 51 possibilities for the second card, 50 for the third and so on. The four of hearts is the last card left, so your final step is to multiply by one.
52! works out as ... brace yourself ... 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000. If you're wondering how to say that, it begins ‘80 unvigintillion ...'
That we know for sure. But how many deck orders have there ever been in human history? Clearly we can only estimate this - but let's make sure it's an over-estimate, to really test the theory. The modern 52-card deck has been around for 500 years or so - let's double that to 1000. And let's say the world population had been at its current level (about 8 billion) for the whole of that time (a huge exaggeration). And now let's say that at any given point, every human being alive had had a deck of cards which they shuffled 100 times a day, every day.
This would have produced, over those 1000 years, a total of 292 quadrillion orders.
So what's the probability that an order you achieve now (one of the 52! possibilities) is one that's been seen before (one of the 292 quadrillion)? You simply divide the first number by the second. This produces an answer with lots of digits in it, but to give you the quotable version - the probability of your order having occurred before is roughly ...
one in 250 trillion trillion trillion trillion.
Of course the slight exception comes when you shuffle a new deck (which starts with all four suits in order) for the first time. If you only shift one or two cards around, that new order might well have cropped up in the past.
But other than that - the next time you shuffle a deck of cards, relish the fact that you've produced something that has almost certainly never been seen before - and will never be seen again.