I understand the math behind it, I just can’t wrap my head around the fact that something so common and limited like a deck of cards can have more ways to be arranged than something so massive like the earth with all its oceans and mountains has atoms.
In my mind it would make more sense that even a little pond has more atoms than there are deck arrangements.
Could it be due to the fact that atoms have a lot of empty space in them?
In: 318
Its not the the number of atoms on earth are unexpectedly small. Its huge, its incomprehensibly gigantic. The issue is that the number of ways to arrange a deck of cards happens to be even more incomprehensibly huge.
It’s confusing because you are comparing completely different things. Count how many dice you can find in your household. Then count how often you can throw the dice. Doesn’t make any sense, right?
The trick and reason this seems counterintuitive is, because we tend to underestimate how quickly the number of ordering rises with each new card.
Two cards have 2 ways of being shuffled. Or 2! of arrangements.
1. (‘A’, ‘B’),
2. (‘B’, ‘A’)
Three cards have 6 ways of being shuffled, 3! arrangements
1. (‘A’, ‘B’, ‘C’), (‘A’, ‘C’, ‘B’)
2. (‘B’, ‘A’, ‘C’), (‘B’, ‘C’, ‘A’),
3. (‘C’, ‘A’, ‘B’), (‘C’, ‘B’, ‘A’)
Four cards have 24 ways of being shuffled 4! arrangements
1. (‘A’, ‘B’, ‘C’, ‘D’), (‘A’, ‘B’, ‘D’, ‘C’), (‘A’, ‘C’, ‘B’, ‘D’), (‘A’, ‘C’, ‘D’, ‘B’)
2. (‘A’, ‘D’, ‘B’, ‘C’), (‘A’, ‘D’, ‘C’, ‘B’), (‘B’, ‘A’, ‘C’, ‘D’), (‘B’, ‘A’, ‘D’, ‘C’)
3. (‘B’, ‘C’, ‘A’, ‘D’), (‘B’, ‘C’, ‘D’, ‘A’), (‘B’, ‘D’, ‘A’, ‘C’), (‘B’, ‘D’, ‘C’, ‘A’)
4. (‘C’, ‘A’, ‘B’, ‘D’), (‘C’, ‘A’, ‘D’, ‘B’), (‘C’, ‘B’, ‘A’, ‘D’), (‘C’, ‘B’, ‘D’, ‘A’)
5. (‘C’, ‘D’, ‘A’, ‘B’), (‘C’, ‘D’, ‘B’, ‘A’), (‘D’, ‘A’, ‘B’, ‘C’), (‘D’, ‘A’, ‘C’, ‘B’)
6. (‘D’, ‘B’, ‘A’, ‘C’), (‘D’, ‘B’, ‘C’, ‘A’), (‘D’, ‘C’, ‘A’, ‘B’), (‘D’, ‘C’, ‘B’, ‘A’)
I will stop making the graphical representations, but at this point I think you can see, that adding a increases the number of combinations by a factor equal to the order of the card (second card doubles the preceding number, third car triples preceding number, fourth quadruples ….)
Fifth card gives us more then hundred combinations
Sextuple that and you get almost a thousand combinations
septuple that and you are half way to ten thousands.
multiplication by 8 gets you almost half way to hundred thousand
9 gets you third of a million
10 gets you 3 millions every successive number therefore increases the preceding number by more than an order of magnitude (add a zero to a number)
18th card, will get you a number of approximately 6 moons worth of atoms
after 20th card (we are now at 2.43×10^18, number of transistors we produced per year in 2008), each new card will increase the order of magnitude and double it. and yet we are not even half way done.
42nd card will get you about number that represent number of atoms on 10 Earths
46th card number of atoms in 5 Suns
47th card would allow you (according to Archimedes) fill a quarter of two light years wide cosmos with grains of sand.
And finally 52 will get you about number that represent number of atoms in 3 milky ways
According to random page I found, a number of atoms in observable universe is somewhere between 10^78 – 10^82 meaning you would need between 58 and 61 cards to have a comparable number of shuffelings
[https://www.youtube.com/watch?v=pUF5esTscZI](https://www.youtube.com/watch?v=pUF5esTscZI) how about doubling the size of a small piece of paper until we cover the universal instead. how many time do you think that would need to happen.
You are counting two completely different things, both of which are hard to grasp intuitively.
First of all, I think that if I told you there are 10^(100) atoms in Earth you would probably believe me. This number is incredibly off. These numbers are so absurdly massive it is difficult to comprehend, but it is the same overestimation as saying “one Earth” when in reality you have one little atom.
Second, the number of permutations is also counterintuitively massive. Have you heard the story of the checkerboard and the grain of rice? A pharaoh was very please with a wise man, and asked how could he pay. The wise man said he wanted the pharaoh to put one grain of rice in the first square of a checkerboard, two in the second one, four in the third, eight in the fourth… Always doubling it. Each four squares the total becomes 16 times larger. By the end of the second round, you are about 1 kg. By the time you hit half the board, you’ve passed one ton. When you fill the board, you hit 100 times the global production of rice last year. [NOTE: my numbers may be off, but the huge growth stands].
The number of shuffles is 52 for 1 card, 52 × 51 = 2652 for 2 cards, 52 × 51 × 50 = 132 600 for 3 cards… For 52 cards, 52 × 51 × … × 2 × 1 is about 10^(68). It is not exactly the same as with the rice, but similar enough to understand it.
It is hard to wrap you head around this, I know. Adding one element makes things explode. For counting atoms in Earth, one atom is just one atom. Twice as many checkerboard squares will (may) make the amount of rice stupidly larger. Twice as many atoms is “just” twice as many atoms.
This is called *exponential growth*. People say a lot “exponentially large”, but they most often use it wrong. It just does not mean “very big”, it means “this grows much, much, MUCH faster the more something else grows”.
Numbers are just numbers. When you use exponents, you can get to very high numbers very fast. The classic example is [trying to fill a chessboard with wheat](https://en.wikipedia.org/wiki/Wheat_and_chessboard_problem). 2^x becomes too high too fast.
You need to stop thinking naively using common (non)sense and start thinking mathematically.
It’s simple enough:
There are 52 cards in a standard deck. That means there are 52 ways to choose the first card.
After you have chosen the first card there are now 51 cards remaining. So there are 51 ways to choose the second card.
That means there are 52 times 51, or 2,652 ways to choose just the first two cards.
Proceeding through the remaining 50 cards, there are 80, 658, 175, 170, 943, 878, 571, 660, 636, 856, 403, 766, 975, 289, 505, 440, 883, 277, 824, 000, 000, 000, 000 or about 8.1 x 10^67 different ways to select 52 cards from a standard deck.
For giggles this an also be expressed as: 2^49 ×3^23 ×5^12 ×7^8 × 11^4× 13^4 ×17^3 ×19^2 ×23^2 ×29 ×31 ×37 ×41 ×43 ×47
There are about 10^50 atoms in the earth.
Counting multiples of things yields larger numbers than does count things.
52!. That is 52*51*50…2*1. So you want to arrange them in a row. You first draw, you have 52 possible cards. On your second draw, you have 51 possible cards to draw. This continues for a total of 52 times, each time with consecutively fewer cards. It’s just a big number that happens to be bigger than the number of atoms in the universe.
We only play with objects, but never with exponentials. Anything that changes exponentially in real life that we can actively estimate is either severely constrained or cut off before the numbers become too big (like a virus taking down a population). So we don’t have a native sense for numbers of combinations.
Because the atoms on earth are actual physical reality. The number of ways to arrange a deck of cards isn’t.
You’re not actually arranging 52! decks of cards in different ways, you’re just thinking about it
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