2022 AMC 10A Problem 9

1. Name the regions. Bottom row, left to right: a small piece $L$, a wide middle piece $M$, and a right piece $R$. Top row: a left piece $U$ and a right piece $V$.
2. Which regions touch? Reading the picture:
- $M$ touches all four others: $L$, $R$, $U$ and $V$.
- $L$ touches $U$; $U$ touches $V$; $V$ touches $R$.
- $L$ does not touch $R$ or $V$, and $U$ does not touch $R$.
3. Paint the busiest region first. $M$ touches everything, so it limits the most. It gets any of the $5$ colors.
4. Go around the other four in the order $L, U, V, R$. Each one touches $M$ and the region painted just before it, and those two already have different colors:
- $L$: any color except $M$'s. $4$ choices.
- $U$: not $M$'s color and not $L$'s. $3$ choices.
- $V$: not $M$'s and not $U$'s. $3$ choices.
- $R$: not $M$'s and not $V$'s. $3$ choices.
5. Multiply. $5 \times 4 \times 3 \times 3 \times 3 = 540$.

The answer is $\boxed{\textbf{(D)}\ 540}$.

When counting colorings, choose an order in which each new region has a known number of already-painted neighbors, all with different colors. Starting with the region that touches everything does this: every later region sees exactly two different colors (or one, for the first), so each count is a fixed number and you simply multiply.