Sure, but in combinatorics the number of atoms in the universe (say 1e80) is not a large number. For example, the factorial of 59 is larger. If you own 30 pairs of shoes, there are factorial(60) ways to arrange the individual shoes in a sequence.
To be fair, that's atoms in the observable universe.
The total size of the universe is unknown, and could (and likely does) have way more atoms than that.
Actually, that's a fun thought: assuming homogenuity of matter between the observable and unobservable universe, how much bigger would the unobservable universe need to be to render some of these claims no longer true?
Because you're right to point out that factorials grow absurdly quickly. It's entirely possible my caveat straight up doesn't matter.
Edit: Ok, I'm seeing Wikipedia has a (disputed) estimate for the diameter of the total universe as 10^10^10^128 megaparsecs. Then, radius cubed should be 1/2(10^10^10^128^3)=1/210^10^10^131, as opposed to the radius of the observable universe being a nice, clean 14 billion parsecs = 1410^3 megaparsecs, making the radius cubed 1410^4 megaparsecs.
I don't think I have a big enough calculator for this, but for fun, let's say 128^3 is roughly 2,000,000. Then we can rewrite T, the relative volume of the total universe, as 1/210^10^10^2*(10^
6). I guess if we call 14 close enough to 10, then our density is 10^80/10^6=10^74 atoms for every pi megaparsecs cubed.
Going off the heuristic that n!<n^n, and the total universe can trivially produce (10^10)^(10^10), we would need to rearrange >10^10 objects just to even start to think about the number of (megaparsecs cubed)/pi it might have, let alone the 10^74 those each have.
We might not have enough decks of cards for this one.
(Feel free to criticize/tear down my math or logic anywhere in this one, it's very much off the cuff and I'm sure I made at least as many egregious errors in computing exponents as I did computations. No math class I've taken yet really prepares you to handle exponents raised four deep.)
BobaFloutist|1 month ago
The total size of the universe is unknown, and could (and likely does) have way more atoms than that.
Actually, that's a fun thought: assuming homogenuity of matter between the observable and unobservable universe, how much bigger would the unobservable universe need to be to render some of these claims no longer true?
Because you're right to point out that factorials grow absurdly quickly. It's entirely possible my caveat straight up doesn't matter.
Edit: Ok, I'm seeing Wikipedia has a (disputed) estimate for the diameter of the total universe as 10^10^10^128 megaparsecs. Then, radius cubed should be 1/2(10^10^10^128^3)=1/210^10^10^131, as opposed to the radius of the observable universe being a nice, clean 14 billion parsecs = 1410^3 megaparsecs, making the radius cubed 1410^4 megaparsecs. I don't think I have a big enough calculator for this, but for fun, let's say 128^3 is roughly 2,000,000. Then we can rewrite T, the relative volume of the total universe, as 1/210^10^10^2*(10^ 6). I guess if we call 14 close enough to 10, then our density is 10^80/10^6=10^74 atoms for every pi megaparsecs cubed.
Going off the heuristic that n!<n^n, and the total universe can trivially produce (10^10)^(10^10), we would need to rearrange >10^10 objects just to even start to think about the number of (megaparsecs cubed)/pi it might have, let alone the 10^74 those each have.
We might not have enough decks of cards for this one.
(Feel free to criticize/tear down my math or logic anywhere in this one, it's very much off the cuff and I'm sure I made at least as many egregious errors in computing exponents as I did computations. No math class I've taken yet really prepares you to handle exponents raised four deep.)