What Derangement is actually computing
A derangement is a permutation with zero fixed points. The recurrence Dₙ=(n−1)(Dₙ₋₁+Dₙ₋₂) builds the count from smaller cases.
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Derangement Calculator
Count permutations in which no object remains in its original position. The calculation trail makes the reported derangements easier to reproduce.
Derangement inputs
Values for this result
Situations modeled by Derangement
It models shuffled assignments where nobody receives their own item, misplaced letters, and complete reassignment constraints.
Building Derangement
Begin with D₀=1 and D₁=0. Apply the recurrence successively until reaching the requested object count.
For seven objects, D₇=1,854. Dividing by 7!=5,040 gives a probability near 0.3679 for a uniformly random permutation. This Derangement example can be compared with all permutations.
Assumptions behind Derangement
The Derangement formula describes a particular experiment built from Distinct object count. Write down whether the trial count is fixed, objects return after selection, events overlap, or outcomes are equally likely. Similar-looking numerical inputs can require different formulas when one assumption changes.
For Derangement, after calculation, translate the answer back into a sentence about the original event. That wording should distinguish exactly, at most, at least, all, and none; substituting one of those phrases for another changes the event rather than its formatting.
Objects and original positions must be distinct. The nearest-integer approximation n!/e is useful for scale but the recurrence provides the exact result. If the Derangement assumptions do not fit, consider derive the restriction.
A second verification of Derangement
Decide whether Derangement orders Distinct object count and allows repetition. Those Derangement choices determine the result.
List a small set of Derangement outcomes. Compare the direct Derangement count with the result.
Before accepting Derangements, restore the Derangement inputs Distinct object count and the fixed condition. Estimate Derangements independently. Then vary the fixed condition alone and observe the new Derangement output. This isolates the changed part of Derangement.
A final Derangement check
A final Derangement note should label Derangements clearly, especially when the number will become an input to another calculation.
Questions about Derangement
What is a fixed point?
An object occupying its original position.
Why is D1 zero?
The only object cannot move elsewhere.
Is n!/e exact?
No, but rounding it gives the exact derangement for positive n.
What does the probability approach?
One divided by e.