Three darts in three different rings

A dartboard is made of three concentric circles of radius $1$, $2$ and $3$, which divide it into an inner disc and two rings. Three darts land independently at uniformly random points on the board. What is the probability that they all land in different regions?

$\tfrac{10}{81}$. The regions have areas proportional to $1$, $3$ and $5$ out of $9$. The darts must occupy them in one of $3!$ assignments, so the probability is $6\cdot\tfrac19\cdot\tfrac39\cdot\tfrac59=\tfrac{90}{729}=\tfrac{10}{81}$.