Difference between revisions of "User:John0512"
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==Orzorzorz Number== | ==Orzorzorz Number== | ||
| − | This is a number I coined. The exact value of this number is <cmath>\sum_{m=17}^{122}\sum_{k=17}^{122}\sum_{n=17}^{\min(m,k)} (2^{24})^{n-17}\cdot(2^{730})^{\min(m,k)-n}\cdot26^{ | + | This is a number I coined. The exact value of this number is <cmath>\sum_{m=17}^{122}\sum_{k=17}^{122}\sum_{n=17}^{\min(m,k)} (2^{24})^{n-17}\cdot(2^{730})^{\min(m,k)-n}\cdot26^{12(m-17)+12(k-17)},</cmath> and is approximately equal to <math>4.8\cdot10^{26639}</math>. The formula used to find this number is an approximation of the number of ways mathimathz can happen, however I will not go into further detail about exactly how it was derived. Each one of the numbers in the formula has a good reason, however I am not revealing just yet. :) |
==Unnamed Theorem== | ==Unnamed Theorem== | ||
Revision as of 10:52, 1 October 2021
Orzorzorz Number
This is a number I coined. The exact value of this number is ![\[\sum_{m=17}^{122}\sum_{k=17}^{122}\sum_{n=17}^{\min(m,k)} (2^{24})^{n-17}\cdot(2^{730})^{\min(m,k)-n}\cdot26^{12(m-17)+12(k-17)},\]](http://latex.artofproblemsolving.com/4/4/8/4485f43e58cfd4b36fe9a054519e9353b34b0a99.png)
and is approximately equal to 
. The formula used to find this number is an approximation of the number of ways mathimathz can happen, however I will not go into further detail about exactly how it was derived. Each one of the numbers in the formula has a good reason, however I am not revealing just yet. :)
Unnamed Theorem
I have something called the Unnamed Theorem (which I did not name as I have not confirmed that this theorem has not existed before).
Claim: Given a set 
where 
is a positive integer, the number of ways to choose a subset of 
then permute said subset is 
Proof: The number of ways to choose a subset of size 
and then permute it is 
. Therefore, the number of ways to choose any subset of is ![\[\sum_{i=0}^n \binom{n}{i}\cdot i! = \sum_{i=0}^n \frac{n!}{(n-i)!}.\]](http://latex.artofproblemsolving.com/4/a/4/4a4de9a20d25a54b46cfe4d264efccfaea7ac703.png)
This is also equal to 
by symmetry across 
. This is also 
Note that 
is defined as 
, so our expression becomes ![\[n!(e-\sum_{i={n+1}}^\infty \frac{1}{i!}).\]](http://latex.artofproblemsolving.com/3/0/a/30ad6f9fbc2afb04fd7f5cce3eb35ef5d148aa31.png)
We claim that 
for all positive integers .
Since the reciprocal of a factorial decreases faster than a geometric series, we have that 
. The right side we can evaluate as 
, which is always less than or equal to 
. This means that the terms being subtracted are always strictly less than , so we can simply write it as ![\[\lfloor n!\cdot e\rfloor.\]](http://latex.artofproblemsolving.com/b/6/b/b6b17898bfb6718898259c7a695b8c08f23af28e.png)
Example: How many ways are there 5 distinct clones of mathicorn to each either accept or reject me, then for me to go through the ones that accepted me in some order?
Solution to example: This is equivalent to the Unnamed Theorem for 
, so our answer is 
.
Solution 2: Since I am not orz, all 5 clones will reject me, so the answer is 
. Note that this contradicts with the answer given by the Unnamed Theorem.
