MathCalcLab

Derangements Calculator

Calculate the number of derangements (permutations with no fixed points) and analyze the subfactorial function.

Enter a number between 0 and 20 to calculate derangements. Derangements are permutations where no element appears in its original position.

Risultati

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Theory & Formula

Derangements Theory

A derangement is a permutation where no element appears in its original position. The number of derangements of n elements is denoted !n (subfactorial of n).

Key Formulas

Series Formula: \(!n = n! \sum_{k=0}^{n} \frac{(-1)^k}{k!}\)
Recursive Formula: \(!n = (n-1)(!(n-1) + !(n-2))\)
Approximation: \(!n \approx \frac{n!}{e}\) for large n

Probability

The probability that a random permutation is a derangement approaches 1/e ≈ 36.8% as n increases.

\(\lim_{n \to \infty} \frac{!n}{n!} = \frac{1}{e} \approx 0.367879\)

Applications

The hat-check problem: n people check their hats, and the hats are returned randomly. What is the probability that no one gets their own hat back? Answer: !n/n! ≈ 1/e.

Example

For n=3: !3 = 2. The permutations [2,3,1] and [3,1,2] are the only derangements of [1,2,3].

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