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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 - dCode

Tag(s) : Combinatorics

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Derangements

Derangements Generator

Counting 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.

Answers to Questions

How to generate derangement?

To generate the list of derangements of a set, the easiest way 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 makes exponential values. The more calculations there are, the more expensive are computer servers, so the large generations must be paid.

Source code

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