15. City $A$ has 4 suburban counties $(B, C, D, E)$, as shown in Figure 2. There are 5 colors available. How many different coloring methods are there such that no two adjacent areas share the same color, and each area is painted with only one color?
15. The coloring method that meets the requirements must use at least three colors, so it can be divided into three categories:
(1) Using five colors, there are $P_{5}^{s}=120$ methods.
(2) Using four colors. The number of ways to choose four colors is $\mathrm{C}_{5}^{4}$. Among them, choosing one color to paint $A$ has $\mathrm{C}_{4}^{1}$ ways, and the remaining 4 blocks are painted with the remaining three colors, with one and only one pair of non-adjacent regions painted the same color. There are 2 ways to choose a pair of non-adjacent regions $(B, D$ or $C, E)$, and choosing one color from the remaining three colors to paint these non-adjacent regions has $\mathrm{C}_{3}^{1}$ ways. Finally, the number of ways to paint the remaining two regions with the remaining two colors is $\mathrm{P}_{2}^{2}$. According to the multiplication principle, there are $C_{5}^{4} \cdot C_{4}^{1} \cdot 2 \cdot C_{3}^{1} \cdot P_{2}^{2}=240$ methods.
(3) Using three colors. The number of ways to choose three colors is $\mathrm{C}_{5}^{3}$. Painting $A, B$ and $D, C$ and $E$ with one color each has $\mathrm{P}_{3}^{3}$ ways, so there are $\mathrm{C}_{5}^{3} \mathrm{P}_{5}^{3}=60$ methods.
According to the addition principle, there are a total of $120+240+60=420$ methods.