Arrows spread evenly over three target zones

A target consists of a central disc of radius 4 inches, surrounded by an annulus out to radius 8 inches, surrounded by another annulus out to radius 12 inches. Every arrow lands at a point uniformly distributed over the whole 12-inch disc, independently.

(a) Three arrows are shot. What is the probability that each of the three zones receives exactly one arrow?

(b) If $3n$ arrows are shot, give a formula for the probability that each zone receives exactly $n$ arrows.

(a) $10/81$; (b) $\dfrac{(3n)!}{(n!)^3}\left(\tfrac{5}{243}\right)^{n}$. Area ratios give zone probabilities $\tfrac19,\tfrac39,\tfrac59$. In (a) the $3!$ orderings each have probability $\tfrac{15}{729}$, so $\tfrac{90}{729}=\tfrac{10}{81}$. In (b) the multinomial coefficient multiplies $\left(\tfrac19\cdot\tfrac39\cdot\tfrac59\right)^n$.