I was going through some problem solving questions, and I came upon this one:
Color this map with the colors red, orange, yellow, green, and blue in such a way that no two adjacent regions have the same colors. In how many ways is this possible?
I tried doing it this way: You can color region A in 5 ways, region B in 4 (because region A already has a color), region C in 4 ways, region D in 2 ways, and region E in 3 ways. That means there are $5(4)(4)(2)(3)=480$ ways to do this.
However, the answer shown is 540. They used the same reasoning as me, but instead of going from A to B to C to D to E, they went from A to B to D to C to E. Why does this make a difference?
In your proposed solution, in the case of B and C having the same color, then region D would have 3 possibilities (instead of 2). With their ordering of filling in the regions, you do not get any different cases as you go.