The subject matter presented in this article can be used in the classroom to enrich the learning experience of students taking a course that includes a unit on combinatorics, such as discrete mathematics, graph theory, or probability. In order to provide such students with the background needed to appreciate the significance of the generalization of "n choose r" described herein, it is recommended that their teachers introduce them to the concept of "n choose r" (indicating its combinatorial significance, its definition in terms of factorials, and its relationship to the Binomial Theorem), the topic of bit strings, and the theory of generating functions. In this article, the author generalizes the concept of "n choose r" and then derives several formulas. This generalization of "n choose r" leads, in a natural way, to a generalization of the notion of Pascal's Triangle. This latter generalization is presented in a separate article that also appears in this issue of Mathematics and Computer Education, entitled "Pascal's Infinite Set of Triangles".