Example 5. In the central square of a city, a flower bed has been built, divided into 6 sections (as shown in the figure). Now, 4 different colors of flowers are to be planted, with one color per section, and adjacent sections cannot have the same color flowers. How many different planting methods are there?
Analysis one: First plant 1, there are $\mathrm{C}_{4}^{1}$ ways, then consider the remaining 5 pieces as a circular ring and cut along one side, which becomes the previous problem (5 pieces 3 colors) and the first and last pieces $2, 6$ cannot be the same color. Subtract the number of ways where $2, 6$ are the same color from the result of the previous problem, and there are $3 \times 2 \times 1 \times 1 + 3 \times 2 \times 1 \times 1 = 12$ ways for $2, 6$ to be the same color, so there are $\mathrm{C}_{4}^{1} \times (42 - 12) = 120$ ways in total.

Analysis two: For 6 pieces and 4 colors, there must be the same color and there will not be 3 pieces of the same color. If we consider the 2 pieces of the same color as one piece, then there are $\mathrm{A}_{4}^{4}$ ways, and the same color situations are: 2, 4 same color and 3, 6 same color, 2, 4 same color and 3, 5 same color, 2, 5 same color and 3, 6 same color, 2, 5 same color and 4, 6 same color, 3, 5 same color and 4, 6 same color, 5 ways in total, so there are $5 \times \mathrm{A}_{4}^{4} = 120$ ways.