Simply put, a derangement is a situation where nothing is in the correct spot!
A simple example would be the ordered letters {A,B,C}. {B,C,A} and {C,A,B} are both derangements because no letter appears in its correct spot.
In detail, a derangement of n ordered objects is a permutation where no object is in its correct position. The number of derangments of n objects is defined by:
It is really that simple. To solve a derangement problem just determine the number of objects which will be the n in your equation and plug the numbers in.
The formula is developed by using the Priciple of Inclusion-Exclusion, and is quite interesting. To see the proof follow the References link below and read the Derangement section of handout 22.
Continue to a real life example
References: Handouts 20 and 22 by Leen-Kait Soh