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 plot showing n! and !n below:
