jmc

We first factor 180 into three positive integers from the set $\{1,2,3,4,5,6\}$. Since $180>5^3=125,$ at least one of the integers must be 6. Since $180>5^2\cdot 6=150$, at least two integers must equal 6. Indeed, $180=5\cdot 6\cdot 6$ is the only such way to factor 180. Therefore, $(a,b,c)=(5,6,6),(6,5,6),(6,6,5)$ are the only possibilities for $a,b,c$. Each occurs with probability ${\left(\frac{1}{6}\right)}^3=\frac{1}{216}$, so the probability that $abc=180$ is $3\cdot \frac{1}{216}=\boxed{\frac{1}{72}}$.