  For a hyperbolic 3-manifold $M$ with a torus boundary component,all but
finitely many Dehn fillings yield hyperbolic 3-manifolds. In this paper, we
will focus on the situation where $M$ has two exceptional Dehn fillings: an
annular filling and a toroidal filling. For such situation, Gordon gave an
upper bound 5 for the distance between such slopes. Furthermore, the distance 4
is realized only by two specific manifolds, and 5 is realized by a single
manifold. These manifolds all have a union of two tori as their boundaries.
Also, there is a manifold with three tori as its boundary which realizes the
distance 3. We show that if the distance is three then the boundary of the
manifold consists of at most three tori.
