We discuss a physically motivated definition for a quantitative measure of the selectivity of electron and hole contacts. We define the selectivity S10 = log10(Vth /(c ×Jc)) to depend on the contact resistance c, the recombination current density Jc of the contact, and the thermal voltage Vth. A high selectivity relies on a highly asymmetric equilibrium carrier concentration of majority and minority carriers in the contact. The maximum efficiency max increases with the selectivity S10. This increase is linear until the efficiency starts to be limited by radiative recombination. We give analytic equations for calculating the maximum efficiency max(S10) of a crystalline Si cell that is ideal except for either one or two contacts. Achieving the maximum efficiency max requires optimized areal fractions fe,max and fh,max for the electron and the hole contacts, respectively . We give analytic equations for these contact fractions.