jmc

First, we can find the denominator of our fraction. There are a total of $\left(\genfrac{}{}{0ex}{}{18}{3}\right)=816$ ways to choose 3 pieces of silverware out of 18. To find the numerator, we need to count the number of ways to choose one piece of each type of silverware. There are 6 ways we could choose a fork, 6 ways to choose a spoon, and 6 ways to choose a knife, making a total of $6\cdot 6\cdot 6=216$ ways to choose a fork, a knife, and a spoon so our final probability is $\frac{216}{816}=\boxed{\frac{9}{34}}$.