Search for a tool
Subfactorial

Tool to compute subfactorial. Subfactorial !n is the number of derangements of n object, ie the number of permutations of n objects in order that no object stands in its original position.

Results

Subfactorial -

Tag(s) : Arithmetics

dCode and you

dCode is free and its tools are a valuable help in games, puzzles and problems to solve every day!
You have a problem, an idea for a project, a specific need and dCode can not (yet) help you? You need custom development? Contact-me!


Team dCode likes feedback and relevant comments; to get an answer give an email (not published). It is thanks to you that dCode has the best Subfactorial tool. Thank you.

Subfactorial

Sponsored ads

SubFactorial Calculator !N


Tool to compute subfactorial. Subfactorial !n is the number of derangements of n object, ie the number of permutations of n objects in order that no object stands in its original position.

Answers to Questions

How to calculate a subfactorial?

SubFactorial \( n \) is a calculated using this formula: $$ !n = n! \sum_{k=0}^n \frac {(-1)^k}{k!} $$

Example: $$ \begin{align} !4 &= 4! ( \frac{(-1)^0}{0!} + \frac{(-1)^1}{1!} + \frac{(-1)^2}{2!} + \frac{(-1)^3}{3!} + \frac{(-1)^4}{4!} ) \\ &= 4! \times ( 1/1 - 1/1 + 1/2 - 1/6 + 1/24 ) \\ &= 24 \times 9/24 \\ &= 9 \end{align} $$

This formula is also used : $$ !n = \left [ \frac {n!}{e} \right ] $$ where brackets [] stands for rounding to the closest integer.

Example: \( 4! / e \approx 24/2.718 \approx 8.829 \Rightarrow !4 = 9 \)

What are the first values of the subfactorial function?

The first values for the first natural numbers are:

!1 = 0
!2 = 1
!3 = 2
!4 = 9
!5 = 44
!6 = 265
!7 = 1854
!8 = 14833
!9 = 133496
!10 = 1334961

How to write a subfactorial?

The subfactorial as the factorial, uses the exclamation mark as symbol but it is written to the left of the number: \( !n \)

How to list derangements

Derangements (or Rencontres) are permutations without the one with fixed points (no item is in its original place). The number of derangements for \( n \) elements is subfactorial of \( n \): \( !n \).

Example: The \( !4 = 9 \) derangements of {1,2,3,4} are {2,1,4,3}, {2,3,4,1}, {2,4,1,3}, {3,1,4,2}, {3,4,1,2}, {3,4,2,1}, {4,1,2,3}, {4,3,1,2}, and {4,3,2,1}.

Ask a new question

Source code

dCode retains ownership of the source code of the script Subfactorial online. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, Matlab, etc.) which dCode owns rights will not be given for free. To download the online Subfactorial script for offline use on PC, iPhone or Android, ask for price quote on contact page !

Questions / Comments


Team dCode likes feedback and relevant comments; to get an answer give an email (not published). It is thanks to you that dCode has the best Subfactorial tool. Thank you.


Source : https://www.dcode.fr/subfactorial
© 2018 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaches. dCode
Feedback