jmc

The graph has a hole at $x=7$ as long as there is at least one factor of $x-7$ in the denominator, but at most five. (If there were more than five, then there would be a vertical asymptote at $x=7.$) Hence, the possible values of $n$ are 1, 2, 3, 4, 5, for a total of $\boxed{5}$ possible values.