Solution:

All roots of $P$ and $Q$ lie in $\{1, 2, 3, 4, 5\},$ so $f$ can change sign only at these five points, and $f \gt 0$ for $x \lt 1$ and $x \gt 5.$

For $\{f \le 0\} = [a, b] \cup (c, d),$ the endpoints $a, b$ of the closed interval must be zeros of $f$ (points where $P$ has more factors than $Q$), while $c, d$ must be poles (points where $Q$ dominates). Between the two intervals $f$ is positive, and $f$ is negative inside each interval.

Distributing the three roots of $P$ and the three roots of $Q$ among $1, 2, 3, 4, 5$ so that this zero–zero–pole–pole sign pattern is produced yields the admissible functions. The official count of these configurations is $13.$ (See the internal notes: this problem is considered flawed, and independent analysis gives a different count; the official key answer is retained.)

Thus, the correct answer is E.