The effect of nonrandomness is introduced in the Holey Huggins hole theory (a modified version of the Simha-Somcynsky theory) based on the quasi-chemical approximation of Guggenheim. The resulting nonrandomness not only is dependent on the energetic interactions but also is determined by the so-called free volume contributions typical of cell and hole theories. The observed influence of free volume is in some respects unexpected. Furthermore, the theory offers an accurate description of experimental equation of state data of polymers and small molecules. A comparison with the random mixing analogue of the present theory shows that the introduction of the quasi-chemical approximation improves the accuracy of the description of experimental data.