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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,Mathematics

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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: $$!n = n! \sum_{k=0}^n \frac {(-1)^k}{k!}$$

You can also use the formula : $$!n = \left [ \frac {n!}{e} \right ]$$ where [] stands for rounding to the closest integer.

### 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.

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}, and so !4 = 9.