jmc

We apply the complement principle: we find the total number of cases in which the 2 green places are adjacent, and subtract from the total number of cases. There are $\frac{10!}{5!2!2!1!}=7560$ ways to arrange the plates in a linear fashion. However, since the plates are arranged in a circle, there are $10$ ways to rotate the plates, and so there are $7560/10=756$ ways to arrange the plates in a circular fashion (consider, for example, fixing the orange plate at the top of the table). If the two green plates are adjacent, we may think of them as a single entity, so that there are now $9$ objects to be placed around the table in a circular fashion. Using the same argument, there are $\frac{9!}{5!2!1!1!}=1512$ ways to arrange the objects in a linear fashion, and $1512/9=168$ ways in a circular fashion. Thus, the answer is $756-168=\boxed{588}$.