This paper constructs higher education admission rules that maximise graduation rates and thus increase the skill level of the work force. An application shows that students with a low level in mathematics in secondary school ought to find it easier to be admitted to an economics programme than to law or psychology programmes, even though economics is the most difficult programme from which to graduate without a strong background in mathematics. Indirect gains from optimal admission include the potential of making whole cohorts of students more able to graduate with a higher education degree.