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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.

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Subfactorial -

Tag(s) : Arithmetics

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# Subfactorial

## 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.

### 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}.

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