5. Given that $P$ is a point inside $\triangle A B C$ and satisfies
$$
2 \overrightarrow{P A}+3 \overrightarrow{P B}+6 \overrightarrow{P C}=0 \text {. }
$$

Determine the ratio of the areas of $\triangle P A B$, $\triangle P B C$, and $\triangle P C A$.
5. 6: 2:3.

As shown in Figure 5, take points $A_{1}, B_{1}, C_{1}$ such that $\overrightarrow{P A_{1}}=2 \overrightarrow{P A}, \overrightarrow{P B_{1}}=3 \overrightarrow{P B}, \overrightarrow{P C_{1}}=6 \overrightarrow{P C}$. Then $\overrightarrow{P A_{1}}+\overrightarrow{P B_{1}}+\overrightarrow{P C_{1}}=\overrightarrow{0}$.
Thus, $P$ is the centroid of $\triangle A_{1} B_{1} C_{1}$.
Therefore, $S_{\triangle P A_{1} B_{1}}=S_{\triangle P B_{1} C_{1}}=S_{\triangle P C_{1} A_{1}}$,
$S_{\triangle P A B}: S_{\triangle P B C}: S_{\triangle P C A}$
$$
=\frac{1}{2 \times 3}: \frac{1}{3 \times 6}: \frac{1}{6 \times 2}=6: 2: 3 \text {. }
$$