User:John0512
Orzorzorz Number
This is a number I coined. The exact value of this number is and is approximately equal to
. It's also approximately
to 210 significant figures. 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
This is also equal to
by symmetry across
. This is also
Note that
is defined as
, so our expression becomes
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
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.