Understanding large-scale patterns in student course enrollment is a problem of great interest to university administrators and educational researchers. Yet important decisions are often made without a good quantitative framework of the process underlying student choices. We propose a probabilistic approach to modelling course enrollment decisions, drawing inspiration from multilabel classification and mixture models. We use ten years of anonymized student transcripts from a large university to construct a Gaussian latent variable model that learns the joint distribution over course enrollments. The models allow for a diverse set of inference queries and robustness to data sparsity. We demonstrate the efficacy of this approach in comparison to others, including deep learning architectures, and demonstrate its ability to infer the underlying student interests that guide enrollment decisions.