Difference between revisions of "Derangement"
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− | A '''derangement''' is a [[permutation]] with no [[fixed point]]s. | + | A '''derangement''' is a [[permutation]] with no [[fixed point]]s. That is, a derangement leaves nothing in its original place. For example, the derangements of <math>(1,2,3)</math> are <math>(2, 3, 1)</math> and <math>(3, 1, 2)</math>. |
+ | The number of derangements of a [[set]] of <math>n</math> objects is sometimes denoted <math>!n</math> and is given by the formula: | ||
+ | <math>\displaystyle !n = n! \sum_{k=0}^{n} \frac{(-1)^k}{k!}</math> | ||
− | <math>\ | + | Thus, the number derangements of a 3-[[element]] set is <math>3! \cdot \sum_{k = 0}^3 \frac{(-1)^k}{k!} = 6\cdot(\frac{1}{1} - \frac{1}{1} + \frac{1}{2} - \frac{1}{6}) = 2</math>, which we know to be correct. |
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Revision as of 17:35, 13 November 2006
A derangement is a permutation with no fixed points. That is, a derangement leaves nothing in its original place. For example, the derangements of are
and
.
The number of derangements of a set of objects is sometimes denoted
and is given by the formula:
Thus, the number derangements of a 3-element set is , which we know to be correct.
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