This paper establishes that every random utility discrete choice model (RUM) has a representation that can be characterized by a choice-probability generating function (CPGF) with speci c properties, and that every function with these specifi c properties is consistent with a RUM. The choice probabilities from the RUM are obtained from the gradient of the CPGF. Mixtures of RUM are characterized by logarithmic mixtures of their associated CPGF. The paper relates CPGF to multivariate extreme value distributions, and reviews and extends methods for constructing generating functions for applications. The results for RUM are extended to competing risk survival models.