Derangements

Derangements

## Definition / Concept

A derangement is a permutation in which no object occupies its original position.

The number of derangements of $$n$$ distinct objects is denoted by:

$$!n$$

## Formula

$$!n=n!\sum_{k=0}^{n}\frac{(-1)^k}{k!}$$

Equivalently:

$$!n=n\cdot!(n-1)+(-1)^n$$

## Usage

- Used in permutation problems where every object must be displaced from its original position.