AMC 12 2025b Problem 13

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. One such coloring is shown below.

/* Made by Mathemagician108 */
unitsize(1.7cm);
pair O = (0, 0);
real R = 2;

pair A = (R, 0);
pair B = (0.5*R, 0.866*R);
pair C = (0, R);
pair D = (-R, 0);
pair E = (-0.954*R, -0.3*R);
pair F = (-0.22*R, -0.975*R);

path sector1 = O -- A .. arc(O,A,B) -- cycle;
path sector2 = O -- B .. arc(O,B,C) -- cycle;
path sector3 = O -- C .. arc(O,C,D) -- cycle;
path sector4 = O -- D .. arc(O,D,E) -- cycle;
path sector5 = O -- E .. arc(O,E,F) -- cycle;
path sector6 = O -- F .. arc(O,F,A) -- cycle;

pen r = rgb(255, 120, 120);
pen g = rgb(120, 255, 120);
pen b = rgb(120, 120, 255);
filldraw(sector1, g);
filldraw(sector2, r);
filldraw(sector3, b);
filldraw(sector4, r);
filldraw(sector5, g);
filldraw(sector6, b);

draw(circle(O, R));

defaultpen(fontsize(12pt));
label("green", (0.57*R, 0.32*R));
label("red", (0.16*R, 0.63*R));
label("blue", (-0.45*R, 0.45*R));
label("red", (-0.69*R, -0.1*R));
label("green", (-0.45*R, -0.5*R));
label("blue", (0.39*R, -0.49*R));

How many different colorings are possible?

\textbf{(A)}~12 \textbf{(B)}~16 \textbf{(C)}~18 \textbf{(D)}~24 \textbf{(E)}~28