We evaluate the dynamical stability of a selection of outer solar system objects in the presence of the proposed new solar system member Planet Nine. We use a Monte Carlo suite of numerical N-body integrations to construct a variety of orbital elements of the new planet and evaluate the dynamical stability of eight trans-Neptunian objects (TNOs) in the presence of Planet Nine. These simulations show that some combinations of orbital elements (a,e) result in Planet Nine acting as a stabilizing influence on the TNOs, which can otherwise be destabilized by interactions with Neptune. These simulations also suggest that some TNOs transition between several different mean-motion resonances during their lifetimes while still retaining approximate apsidal antialignment with Planet Nine. This behavior suggests that remaining in one particular orbit is not a requirement for orbital stability. As one product of our simulations, we present an a posteriori probability distribution for the semimajor axis and eccentricity of the proposed Planet Nine based on TNO stability. This result thus provides additional evidence that supports the existence of this proposed planet. We also predict that TNOs can be grouped into multiple populations of objects that interact with Planet Nine in different ways: one population may contain objects like Sedna and 2012 VP113, which do not migrate significantly in semimajor axis in the presence of Planet Nine and tend to stay in the same resonance; another population may contain objects like 2007 TG422 and 2013 RF98, which may both migrate and transition between different resonances.