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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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,CB,A,CC,A,BA,C,BB,C,AC,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 3 remaining permutationsC,A,BB,C,A and C,B,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.

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