Counting problems have applications in probability and computer science, and they provide rich contexts for problem solving. Such problems are accessible to students, but subtleties can arise that make them surprisingly difficult to solve. In this paper, students' work on the Groups of Students problem is presented, and an important issue related to multiplication (a fundamental aspect of counting) is discussed. Examples from two categories of students are presented--those who were able to make sense of a correct solution to the counting problem, and those who were not. Evidence is provided that many students failed to attend to the implied order in the multiplicative process, and a productive way of thinking that emerged for some students is also shared. Pedagogical implications and suggested avenues for further work are provided. [For the complete proceedings, see ED584443.]