The odds of a shuffled deck

There are more ways to arrange a deck of cards than there are atoms in our Solar System. In every hand ever dealt, almost certainly no two true shuffles have ever been the same. Almost certainly none ever will be.

How many ways can 52 cards fall?

Start with just a handful. Three cards can fall six different ways. Ten cards can fall 3,628,800 ways. Every card you add multiplies everything again - so the number climbs fast. Shuffle a full deck of fifty-two properly and the count of possible arrangements is 52 factorial, written 52! - and it looks like this:

Sixty-eight digits: about 8.07 × 10⁶⁷ possible arrangements - more than there are atoms in our entire Solar System. Deal one new arrangement every second since the Big Bang and you would still have covered less than a trillionth of a trillionth of a percent of them.

A way to imagine it

Numbers this big stop meaning anything, so here's a way you can try to imagine it:

1. Take a stopwatch, and set it for 52! seconds - that's one second for each possible unique shuffle.
2. Find yourself a position anywhere on Earth's equator, and start the stopwatch. Now, you wait - for one billion years.
3. Once the time has elapsed, take a single step forward, and then wait another billion years.
4. Keep doing this, taking one step every billion years until you've looped around the entire Earth and returned to your starting point.
5. Once you've arrived, take a single drop of water out of the Pacific Ocean. Set it aside, and start the process all over again.
6. Each time you lap the Earth, take another single drop, repeating until the Pacific Ocean is empty.
7. Once the ocean is empty, lay a single sheet of paper on the ground beside you. Refill the ocean, and start again.
8. Continue this entire procedure - one step every billion years, one drop for each lap of the Earth, one sheet every time the ocean is emptied.
9. Once you've done this enough times that the stack of paper is tall enough to reach the Sun, look at your stopwatch; the first 3 digits still haven't changed.
10. Remove the stack of paper and do it over and over again. Once your stack of paper has reached the Sun for the 1000th time, the countdown will only be a third of the way done.

Adapted from Scott Czepiel's essay Understanding the monster that is 52 factorial.

Has a deck of cards ever been shuffled the same way twice?

Almost certainly not. Any two properly shuffled decks are two random draws from those 8.07 × 10⁶⁷ arrangements, and even counting every shuffle by every card player in history, the chance that any two of them have ever collided is vanishingly small. The order of cards a true shuffle creates in your hands has almost certainly never existed, and almost certainly never will again. A one-time event in the history of the universe.

Two shuffled decks can, however, come close. Between any two random decks, the expected number of cards that land in the same position is exactly one. Seven cards in the same places is roughly a 1-in-13,700 event for a given pair; a perfect fifty-two is the 1-in-52! event that has, almost certainly, never happened. The per-pair odds for every count are on the how it works page.