At times, a reliability analyst is faced with life data where failures are found only by inspection. Test items must be inspected at the end of a test period, and then classed as failure or non-failure. Individual failure times are not known. When each item is inspected at a common age, the binomial distribution is routinely applied in practice and in the literature. This paper uses combinatorial mathematics to apply the binomial distribution to the more complicated case where each item is inspected at an arbitrary age. Conditional probabilities are used to extend the analysis when items undergo intermediate inspections.