Formulae of the free energy and the radial distribution function in the hyper-netted chain approximation are obtained for multi-component systems. The expansion formulae by means of the hyper-netted chains are also given for them, and also for the potentials of averate force of n particles. Another expansion scheme is given which is suitable for comparison with the theories of ionic solutions in the past. The ring and watermelon approximations in this scheme are discussed in detail for the Coulombic potential and the Coulombic potential with hard core.