The critical behavior of the mixed Ising model of the type AB(p)C(1-p) ternary alloy consisting of spins sigma = 1/2, S = 1, and m = 3/2 is investigated on the Bethe lattice by using the exact recursion relations. The exact expressions for the magnetizations and magnetic susceptibilities are found, and the thermal behaviors of the magnetizations and susceptibilities are studied. The magnetizations and susceptibilities have also been investigated as functions of the crystal-field interaction or single-ion anisotropy. The phase diagram has been constructed according to which the system always undergoes a second-order phase transition for the coordination number q <= 3 and second- and first-order phase transitions for q > 3; hence, the system has a tricritical point. The system also exhibits reentrant behaviors.