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
You’re comparing a number of things with the number of ways to arrange some things, they’re different categories.
Counting things seems to involve large numbers, but moving around the order of things counterintuitively involves much larger numbers.
For example, I can count six things pretty quickly, but I can arrange those six things 720 different ways.
If I double the six things to 12, I can arrange them nearly half a billion ways.
As you get more things, the counting goes up in a linear way. But the sorting goes up way, way faster. Resulting in a situation where 52 things can be arranged in more ways than all the atoms in the Earth can be counted.
It’s in fact a mind trick.
The number of atoms in the galaxy/universe/whatever is a large, however finite number. There are way more numbers existing.
The thing with the card deck combination is that they don’t exist. All of those decks are just imaginary things, not real objects. Some of them do exist of course but not all.
What’s happening is that when you compare the large count of atoms to a large number of possibilities, your brain puts it into a context of counting real things. So basically your brain tries to imagine a different deck of card near to each atom, but because the deck itself is made of atoms, so it’s impossible.
> I understand the math behind it
Respectfully, if you did you wouldn’t be asking this question.
Because volumes grow multiplicatively, but combinations grow exponentially.
If you make a thing twice as long it becomes heavier by some factor. If you add an extra card, you add to the exponent.
But all the arrangements are ‘fake’. They only exist in theory.
Earth is just sitting there, atoms and all.
Instead of deck of cards, I prefer to use “paragraphs of text in English”.
Take a page of a book. What is numbers of all the possible combination of sentences (grammatically correct, that make some sense) that could be written on that page? Already more than the number of atoms on earth.
The number of things you can describe is absurdly high. It’s like all the alternate universe that could be, which is significantly more than what a single universe has to offer.
And that’s the same thing with a deck of card. Sure, the deck of card looks small, but we’re talking about all the alternate universe where the deck of card is in a different order. That’s a lot. On the other hand, you’re only counting the atoms of the earth of a single universe.
If you were to count the atoms of all the alternate universes too, obviously it would be more than the number of permutations of cards. But outside of extreme cases, counting the things “in one universe” will lead to smaller numbers than counting the things “in all possible universes”.
And that’s because if you add a single object in the universe, the number of possible universes increases exponentially. In the same way that if you add a single word to English, that word can be combined in all the other words to craft an immense amount of new potential sentences.
So if you add enough objects, “all possible universes” will always win over “the content of one universe”.
Lets do an estimation. If we know the Earth’s volume Ve and the volume of an atom Va we can estimate how many of them are roughly in Earth by Ve/Va. Ve = ~ 10²¹ m³ and Va= ~ 4/6 (10^(-10))³ pi this gives us for Ve/Va = ~ 4.77 × 10⁵⁰ number of atoms.
How many ways can 52 different cards can be arranged. Well you have 52 options for the first card, 51 for the second 50 for the third and so on. We multiply the number of possibilities for each card in the deck since any card can be second except for that 1 which is first. So the total number of psooible arrangements or rather permutations is 52! = 52×51×50×49×…×3×2×1. Now 52! = ~8×10⁶⁷ possible permutations.
So the number of atoms in Earth are smaller than all possible permutations of 52 cards. Our estimated value for the number of atoms in Earth is about 42!/3. So even 42 cards can be arranged in more ways than the number of atoms.
There are some great answers already, that count things, but let me put it this intuitive way…
Imagine that you are given the task to give a name to each atom on earth. They line up waiting for their turn and come to you one by one. And you decide that the name you are going to each atom is a particular configuration of the deck of cards. So, each atom shows up and you say “I hereby declare that your name is going to be [particular sequence]”, and the next one show up, etc. It’s just that a standard deck of card is enough to give a unique name to each atom. That’s because although there are lots of atoms, there are even more possible ways to shuffle a deck of card.
In particular that has nothing to do with the distance between atoms, just how many of them there are.
I think your problem is grasping how things that are physically small can yield extremely large numbers. There is some intuition thinkable that any concept of size has to be constrained to some extend by its physical size.
What you have to intuit is that permutation grow extremely fast and that many things in nature that are large are not large due to permutations but rather sheer numbers.
There are not many obvious things in nature that are large due to the effects of permutations and so you are not used to things scaling up like that. A notable example where it does happen is entropy. The fact that for example gasses diffuse through a room is not itself a law of nature. Its pure statistics and in fact the probabilities are so enormous that diffusion MUST happen almost by statistical exclusion.
When it comes to combinatorics, the math of all the possible ways something could exist, the number of atoms in the universe just isn’t that big of a number because it represents the sum of one of those possible combinations.
There’s more ways to order at Chipotle than there are atoms in the universe.
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