jmc

Let us consider the number of three-digit integers that do not contain $3$ and $5$ as digits; let this set be $S$. For any such number, there would be $7$ possible choices for the hundreds digit (excluding $0,3$, and $5$), and $8$ possible choices for each of the tens and ones digits. Thus, there are $7\cdot 8\cdot 8=448$ three-digit integers without a $3$ or $5$.

Now, we count the number of three-digit integers that just do not contain a $5$ as a digit; let this set be $T$. There would be $8$ possible choices for the hundreds digit, and $9$ for each of the others, giving $8\cdot 9\cdot 9=648$. By the complementary principle, the set of three-digit integers with at least one $3$ and no $5$s is the number of integers in $T$ but not $S$. There are $648-448=\boxed{200}$ such numbers.