This study investigates students’ concept image of \(0!\) among 12th- and 13th-grade students as well as second-semester university students by asking them to explain the value of \(0!\) . The analysis of the responses revealed two different categories of explanations: mathematical explanations based on the empty product or on reasons of consistency and the supposed application of the factorial formula, as well as explanations based on the characteristics of 0. The latter type—mainly using multiplication and exponentiation—were predominant among school and university students. In line with the theoretical emphasis on the epistemic role of definitions, this study highlights that definitions serve not only to ensure mathematical consistency but can also acquire specific meanings that foster the application of mathematical concepts. Developing a suitable concept image of \(0!\) , aligned with its concept definition, is essential already at school in order to support correct application in combinatorics and probability theory. Overall, the study shows that both school and university students experience difficulties providing an adequate explanation for \(0!=1\) . This can lead to further complications if the value of \(0!\) is not discussed in class. For example, students cannot make the connection between the two formulas for arrangement and permutation, perceiving \(0!=1\) as an arbitrary determination that seems counterintuitive at first glance. In contrast, when students are familiar with the underlying reasoning, they can develop a deeper conceptual understanding rather than relying solely on a mechanical application of the rules.
