Mathematics in Everyday Life
So, what makes 52 cards so special?
A standard deck has 52 cards. That sounds ordinary. But the number of ways those 52 cards can be arranged is not ordinary at all. It is one of the largest numbers you will ever encounter outside of physics textbooks.
That number is called 52 factorial, written as 52! and understanding it changes how you see the deck forever.
Why 52 × 51 × 50 × ... ?
Think of it like filling seats. You have 52 cards and 52 positions to place them in. Each time you place a card, your remaining choices shrink by one.
First position
All 52 cards are available. You pick one.
Second position
One card is already placed. 51 remain.
Third position
Two placed. 50 left to choose from.
This continues all the way down to the last card
Last position
Only 1 card left. No choice remains.
Multiply them all together
52 × 51 × 50 × 49 × ... × 2 × 1
That product is the total number of possible arrangements. And it equals:
80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000
Roughly 8 × 10⁶⁷. A number with 68 digits.
How big is that, really?
Numbers with 68 digits mean nothing to the human brain. So here is a thought experiment, popularised by mathematician Scott Czepiel, that tries to make it real.
The thought experiment
The Equator Walk
Set a timer for seconds. Stand on the equator and take one step every billion years. After walking all the way around the Earth, take one drop of water out of the Pacific Ocean.
The Pacific Ocean
Repeat the walk until the entire Pacific Ocean is empty. Once empty, place a single sheet of paper on the ground, refill the ocean, and start over.
The Paper Stack
Continue this until your stack of paper reaches the Sun. Even after completing this entire process, the timer will have barely changed.
Putting it next to other big numbers
| What is being counted | Approximate number |
|---|---|
| Seconds since the Big Bang | 4 × 10¹⁷ |
| Atoms in a grain of sand | 10¹⁹ |
| Atoms in the observable universe | 10⁸⁰ |
| Ways to arrange a deck of cards | 8 × 10⁶⁷ |
Why every shuffle is unique
Because 52! is so large, the chance of two properly randomised shuffles ever landing on the same arrangement is effectively zero.
Every casino game, every poker night, every quiet game of solitaire has almost certainly produced an order of cards that had never existed before and will never appear again. It may sound absurd, but this is simply what happens when the number of possible arrangements becomes unimaginably large.
This same math shows up in three places you use every day
Your bank password
Encryption works by making the space of possible keys so enormous that guessing the right one is computationally impossible. Same principle as a shuffled deck, just applied to data security.
Your DNA
Your genome contains around 3 billion base pairs. The number of possible human genetic combinations is so large that no two humans who have ever lived are genetically identical, except identical twins. Combinatorics is why you are unique.