User:John0512
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
.