Color options for a flag

Question

A flag has 3 horizontal stripes with a vertical bar on either side. With 10 different colors how many different ways are there to color this five region flag with no two touching regions the same. I get $(10\cdot(9\cdot8\cdot9))+(10\cdot9\cdot(8\cdot7\cdot9))=46800$ but I am told the right answer is $41040$. Have I gone wrong?

Answer 1

You can have the following configurations of the flag:

$$\left[\begin{array}{c|c|c} & A & \\ C & B & C \\ & A &\end{array}\right]$$ $$\left[\begin{array}{c|c|c} & A & \\ C & B & D \\ & A &\end{array}\right]$$ $$\left[\begin{array}{c|c|c} & A & \\ D & B & D \\ & C &\end{array}\right]$$ $$\left[\begin{array}{c|c|c} & A & \\ D & B & E \\ & C &\end{array}\right]$$

where $A, B, C, D, E$ are different colours.

For each of those, we need to choose, respectively, $3, 4, 4$ or $5$ colours out of $10$. Thus, the result is $(10\cdot 9\cdot 8)+2(10\cdot 9\cdot 8\cdot 7)+(10\cdot 9\cdot 8\cdot 7\cdot 6)=41040$