AMC 10 2022a Problem 9

A rectangle is partitioned into $5$ regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?

$\textbf{(A) }120 \textbf{(B) }270 \textbf{(C) }360 \textbf{(D) }540 \textbf{(E) }720$