For each set $I\subseteq\{1,2,3,4,5\}$ with $|I|=k$ we have $|P_I|=(5-k)!$.
The number of such subsets is
$ \left(\begin{matrix}5\\ k\end{matrix}\right) = \frac{5!}{k!(5-k)!}$,
so
$$ \sum_I |P_I| = \frac{5!}{k!(5-k)!} (5-k)! = \frac{5!}{k!}. $$
Thus, the inclusion-exclusion principle gives
$$ |D| = \sum_k (-1)^k \frac{5!}{k!}
= 5!\left( \frac{1}{0!} - \frac{1}{1!} + \frac{1}{2!}
- \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!}
\right) = 120 - 120 + 60 - 20 + 5 - 1 = 44.
$$