A *derangement* (or *complete permutation*) of a
set is a permutation that leaves no
element in its original position. The number of derangements of a
set with *n* elements can be computed recursively using this
formula:

*D*(*n* + 1) =
*n* (*D*(*n*) + *D*(*n*–1))

Using the principle of inclusion and exclusion, we also get:

Here are some diagrams that represent the derangements of sets
with *n* elements.

3 elements, 2 derangements:

4 elements, 9 derangements:

5 elements, 44 derangements:

The counts are also known as *subfactorials*, with notation !*n*.

Derangements also count *n*×*n* chessboards with *n* non-attacking rooks that avoid the main (NW–SE) diagonal.

*D*_{3} = 2:

*D*_{4} = 9:

*D*_{5} = 44:

Derangement formulas from Fred S. Roberts, Applied Combinatorics, Prentice-Hall, 1984.

Designed and rendered on various weekends over the years using *Mathematica*
versions 3.0, 6.0, 7.0, and 13.2.

© 1996–2024 by Robert Dickau.

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