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Derangements

Tool for generating derangements. In mathematics, a derangement is a permutation of distinct objects without fixed point, ie that no object is in its original position.

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Derangements -

Tag(s) : Combinatorics

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# Derangements

## Counting Derangements

### What is a derangement? (Definition)

The disturbances associated with a set of elements are a subset of its permutations. A fault is a permutation of elements without fixed points, that is to say without elements in the same position with the starting position of the whole.

Example: The set A,B,C has 2 faults C,A,B and B,C,A.

### How to generate derangement?

The faster way to generate the list of derangements of a set is to list the permutations and remove those with fixed points (elements having an identical position in the permutation and in the starting position).

Example: The set A,B,C has 6 permutations: A,B,C B,A,C C,A,B A,C,B B,C,A C,B,A. Remove the one with fixed points, ie. the permutations with A in position 1, and/or those with B in position 2 and/or those with C in position 3.
The list of derangements are the 2 remaining permutations C,A,B and B,C,A.

### How to count derangement?

Counting derangements uses subfactorials. For $n$ items, the number of derangements is equal to $!n$ (subfactorial of $n$): $$!n = n! \sum_{k=0}^n \frac {(-1)^k}{k!}$$

### How to remove the limit when computing derangements?

Derangements make exponential values. The more calculations there are, the more expensive are computer servers, so the large generations must be paid.

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Derangements on dCode.fr [online website], retrieved on 2022-09-30, https://www.dcode.fr/derangements-generator

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