Generating questions based on answers and relevant contexts is a challenging task. Recent work mainly pays attention to the quality of a single generated question. However, question generation is actually a one-to-many problem, as it is possible to raise questions with different focuses on contexts and various means of expression. In this paper, we explore the diversity of question generation and come up with methods from these two aspects. Specifically, we relate contextual focuses with content selectors, which are modeled by a continuous latent variable with the technique of conditional variational auto-encoder (CVAE). In the realization of CVAE, a multimodal prior distribution is adopted to allow for more diverse content selectors. To take into account various means of expression, question types are explicitly modeled and a diversity-promoting algorithm is proposed further. Experimental results on public datasets show that our proposed method can significantly improve the diversity of generated questions, especially from the perspective of using different question types. Overall, our proposed method achieves a better trade-off between generation quality and diversity compared with existing approaches.