let me start by defining a function $D(n)$ for Derangement of $n$ distinct items.
if there are n distinct items, they can be arranged in $n!$ different ways.
so , $$ D(n) = n! - (\binom{n}{1}D(n-1) + \binom{n}{2}D(n-2) + \binom{n}{3}D(n-3) + .... + \binom{n}{n-n}D(n-n))$$
I have substracted all number of ways in which 1 , 2 , 3 ... ,n items are on its place and no other item is on its place.
is there any way to solve it further solve this equation and get it as a function of n only ?
