In A Nutshell
There are more possible arrangements in a deck of cards than there are stars in the known universe. The full number is 52 factorial, which is (very, very roughly) an eight followed by 67 zeroes.
The Whole Bushel
The standard 52-card deck has been around for 500 years or more. What is rather unbelievable is that there are so many possible arrangements of the cards, it is statistically unlikely that any two have ever repeated in all of history. There are in fact 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000 arrangements.
The above number, while impossible to get your head around in any conventional manner, is 52 factorial, the possible orders of a shuffled deck. While enormous, the number is rather simply arrived at, by multiplying 52 down to 1 (52 x 51 x 50 . . . ). There are more possible arrangements of cards than there are atoms on Earth.
With such an astonishing range, the science of “card counting,” or predicting the next cards to be dealt in a hand, seems more magic than math. There are, however, well-known formulas to tip the odds in a player’s favor. In American casinos, card counting is not illegal, though casinos frown upon it and employ various countermeasures to keep professional players from raking in huge pots.
Show Me The Proof
True Count Calculation: The Whole Story
Shuffling Into Hyperspace