The Derangement Problem

A permutation of $\{1, 2, \ldots, n\}$ is called a derangement if no element appears in its original position. For example, $(2, 1, 3)$ is not a derangement of $\{1, 2, 3\}$ because $3$ is in its original position, but $(2, 3, 1)$ is a derangement.

Let $D_n$ denote the number of derangements of $\{1, 2, \ldots, n\}$.

1. Using the Inclusion-Exclusion Principle, derive a closed-form formula for $D_n$.
2. what is the probability that a uniformly random permutation of $\{1, 2, \ldots, n\}$ is a derangement ?