Example 3. A region is divided into five administrative areas as shown in the figure. There are 4 colors available to color the map, with the requirement that adjacent areas must not use the same color. How many different coloring methods are there?
Analysis one: Starting from the 1st position, 1, 2, and 3 have 4, 3, and 2 ways of coloring, respectively. (1) If 4 and 2 are the same color, then 4 has only 1 way of coloring, and 5 has 2 ways, i.e., $4 \times 3 \times 2 \times 1 \times 2 = 48$ ways; (2) If 4 and 2 are different colors, then 4 has 1 way, and 5 also has 1 way, i.e., $4 \times 3 \times 2 \times 1 \times 1 = 24$ ways. In total, there are $48 + 24 = 72$ ways of coloring.

Analysis two: 5 regions with 4 colors, at least two regions must be the same color.
(1) If only 3 and 5 are the same color, there are $\mathrm{C}_{4}^{1}$ ways of coloring, and 1, 2, 4 being different colors have $\mathrm{A}_{3}^{3}$ ways of coloring, totaling $\mathrm{C}_{4}^{1} \cdot \mathrm{A}_{3}^{3} = 24$ ways of coloring;
(2) If only 2 and 4 are the same color, there are $\mathrm{C}_{4}^{1}$ ways of coloring, and 1, 3, 5 being different colors have $\mathrm{A}_{3}^{3}$ ways of coloring, totaling $\mathrm{C}_{4}^{1} \cdot \mathrm{A}_{3}^{3} = 24$ ways of coloring;
(3) If 2 and 4 are the same color and 3 and 5 are the same color, there are $\mathrm{A}_{4}^{2}$ ways of coloring, and 1 has $\mathrm{C}_{2}^{1}$ ways of coloring, totaling $\mathrm{A}_{4}^{2} \cdot \mathrm{C}_{2}^{1} = 24$ ways of coloring.
In summary, there are $24 + 24 + 24 = 72$ ways of coloring.