Textbook Problem

A derangement of the set {1,2,….,n} is a permutation that moves every element of the setaway from of {1,2}, and 231 and 312 are derangements of {1,2,3}. For each positive integers n, let $d_n$ be the number of derangements of the set {1,2,….,n}.

(a) Find $d_1,d_2$ and $d_3$ .
(b) Find $d_4$

Find a recurrence relation for $d_1,d_2,d_3,\mathrm{...}$