The extension of random copolymer Flory-Huggins (FH) type theories to compressible systems is applied to describe the miscibilities of A(x)B(1-x)/C = A,B and A(x)B(1-x)/A(y)B(1-y) binary mixtures. The compressible FH type treatment is also generalized to treat lower critical solution temperature phase diagrams. The latter modification involves a model for the leading contribution to the noncombinatorial entropy based on results from the lattice cluster theory for A/B homopolymer blends in which the monomers are assigned specific molecular structures. The blend miscibility gaps, pressure dependence, critical temperature variations, etc. are analyzed and compared with experimental data. In spite of the several noted limitations, the simple theoretical approach qualitatively explains a number of general trends. A more sophisticated molecular theory, which accounts for monomer sequence dependence of the free energy is, however, necessary for the detailed understanding of random copolymer miscibilities.