We have derived a statistical model of zinc-blende ternary group-III nitrides, assuming the crystal lattice distortion due to lattice constant mismatch to make the major contribution to the free energy of mixing of these compounds and using a new methodology based on the numerical calculation of the configuration partition functions of the alloys. As a result, we have found a nonlinear compositional dependence of the interaction parameter for each of the ternary alloys, $\mathrm{AlGaN}$, $\mathrm{InGaN}$, and $\mathrm{AlInN}$, as well as the temperature-dependent contribution to its free energy originated from the configurational dispersion of the mixing energy. The valence-force field simulations made with the Keating potential have provided the model parameters for these compounds. A simple analytical approximation applicable in the whole range of composition variation is suggested for the interaction parameters of the ternary alloys. The theoretical predictions are compared with those obtained by other approaches and with the data on conventional III-V compounds.