Deriving Derangement Theorem

Deriving Derangement Theorem

The growth of both the functions n! (factorial) and !n (derangement) is exponential, look at the table of values below:

n	Permutation	Derangement
2	2	1
3	6	2
4	24	9
5	120	44
6	720	265
7	5040	1854

We will work with the log (base Math.E) of these functions. Look at the table of values below:

n	log(Permutation)	log(Derangement)
2	0.693	0
3	1.791	0.693
4	3.178	2.197
5	4.787	3.784
6	6.5792	5.5797
7	8.5251	7.5251

We see the following relationship between these values:
log(!n) = log(n!) - 1
=> log(!n) = log(n!) - log(e)
=> log(!n) = log(n! / e)
=> !n = n! / e
And true relationship between !n and n! is: !n = round(n! / e)

Test this out by adding data to the logarithmic plot showing the relationship between n! and !n below:

Tags: Mathematical Foundations for Data Science,