A circle has been divided into 6 sectors of different sizes. Then 2 of the sectors are painted red, 2 painted green, and 2 painted blue so that no two neighboring sectors are painted the same color. How many different colorings are possible?
$$\textbf{(A) } 12 \qquad \textbf{(B) } 16 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 24 \qquad \textbf{(E) } 28$$
Difficulty: 42/100 - A silliable counting problem which is approachable for all.
Core Concepts: counting, casework, coloring
Challenges: Less experienced competitors may go on the approach of counting every possible case rather than first setting two colors and then counting.
Techniques: Apply casework in pairs of colors.
Error-prone Steps: Many competitors miscounted the ways of assigning colors after the first 2 were assigned, or forgetting/messing up a case.
Ideal Time:
Experienced: ≤ 1-2 min
Intermediate: ≤ 2-4 min
Beginner: ≤ 6 min