2015 AMC 10B Problem 16

Let $(A,B,C)$ be the numbers assigned to Al, Bill, and Cal. We need $A$ to be a multiple of $B$, and $B$ to be a multiple of $C$, with all three numbers distinct. The valid triples are $$\begin{gathered} (4,2,1),(6,2,1),(8,2,1), \\ (10,2,1),(6,3,1),(9,3,1), \\ (8,4,1),(10,5,1),(8,4,2). \end{gathered}$$ There are $9$ favorable assignments. The total number of assignments is $10\cdot9\cdot8=720$, so the probability is $\frac9{720}=\frac1{80}$. Thus, the correct answer is C.