Combinatorics is a growing topic in mathematics with widespread applications in a variety of fields. Because of this, it has become increasingly prominent in both K-12 and undergraduate curricula. There is a clear need in mathematics education for studies that address cognitive and pedagogical issues surrounding combinatorics, particularly related to students' conceptions of combinatorial ideas. In this study, I describe my investigation of students' thinking as it relates to counting problems. I interviewed a number of post-secondary students as they solved a variety of combinatorial tasks, and through the analysis of this data I defined and elaborated a construct that I call set-oriented thinking. I describe and categorize ways in which students used set-oriented thinking in their counting, and I put forth a model for relationships between the formulas/expressions, the counting processes, and the sets of outcomes that are involved in students' counting activity.