Difference between revisions of "User:John0512"
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Since the reciprocal of a factorial decreases faster than a geometric series, we have that <math>\sum_{i={n+1}}^\infty \frac{1}{i!}<\frac{1}{(n+1)!}+\frac{1}{(n+1)!(n+1)}+\frac{1}{(n+1)!(n+1)^2}\cdots</math>. The right side we can evaluate as <math>\frac{1}{n(n!)}</math>, which is always less than or equal to <math>\frac{1}{n!}</math>. This means that the terms being subtracted are always strictly less than <math>\frac{1}{n!}</math>, so we can simply write it as <cmath>\lfloor n!\cdot e\rfloor.</cmath> | Since the reciprocal of a factorial decreases faster than a geometric series, we have that <math>\sum_{i={n+1}}^\infty \frac{1}{i!}<\frac{1}{(n+1)!}+\frac{1}{(n+1)!(n+1)}+\frac{1}{(n+1)!(n+1)^2}\cdots</math>. The right side we can evaluate as <math>\frac{1}{n(n!)}</math>, which is always less than or equal to <math>\frac{1}{n!}</math>. This means that the terms being subtracted are always strictly less than <math>\frac{1}{n!}</math>, so we can simply write it as <cmath>\lfloor n!\cdot e\rfloor.</cmath> | ||
− | Example: How many ways are there 5 clones of mathicorn to each either accept or reject me, then for me to go through the ones that accepted me in some order? | + | 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 <math>n=5</math>, so our answer is <math>\lfloor 120e \rfloor=\boxed{326}</math>. | Solution to example: This is equivalent to the Unnamed Theorem for <math>n=5</math>, so our answer is <math>\lfloor 120e \rfloor=\boxed{326}</math>. |
Revision as of 23:40, 14 September 2021
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
.