52!

It’s pretty much guaranteed that no two decks of cards have been shuffled randomly in the same order on earth before.

Our intuition stops working past a certain scale. We can’t really grasp how big these numbers are.

The total number of permutations of a deck of cards is \(52!\)

\(8 \times 10^{67}\)

80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000

One way of trying to feel the size of this number is to do the following:

Imagine you’re standing on earth. Wait a billion years. Then take a step. Wait another billion years. Then take another step. Do that until you’ve walked around the world. Then take a drop of water out of the Pacific Ocean. Then walk around the world again, waiting a billion years in between each step, and take another drop of water out of the Pacific Ocean. Keep doing this until you’ve emptied the Pacific Ocean. Then put a sheet of paper on the floor. Refill the ocean and do the same thing again, each time waiting a billion years in between each step, and when you empty the Pacific Ocean again add another sheet of paper to the pile. When the pile of papers reach the sun… burn the stack of paper and start again… doing the whole thing 1,000 times.

Each second of all that is a possible permutation of a deck of cards.