Derangements
If n distinct items are arranged, the number of ways they can be arranged so that they do not occupy their intended spot is $$D = n!$$($$ \dfrac{1}{0!}$$ - $$\dfrac{1}{1!}$$ + $$\dfrac{1}{2!}$$ - $$\dfrac{1}{3!}$$ + .... + $$\dfrac{(-1)^{n}}{n!}$$)
For, example, Derangements of 4 will be D(4) = $$4!\left(1-\dfrac{1}{1!}+\dfrac{1}{2!}-\dfrac{1}{3!}+\dfrac{1}{4!}\right)=24\left(\dfrac{1}{2}-\dfrac{1}{6}+\dfrac{1}{24}\right)=24\left(\dfrac{12-4+1}{24}\right)=9$$
D(1) = 0, D(2) = 1, D(3) = 2, D(4) = 9, D(5) = 44, and D(6) = 265
