In this work, the statistical distribution functions for boson, fermions and their mixtures have been derived and it is found that distribution functions follow the symmetry features of $\beta$ distribution. If occupation index is greater than unity, then it is easy in the present approach to visualise condensations in terms of intermediate values of mixing parameters. There are some applications of intermediate values of mixing parameters.

There is a need to compute and work out the theoretical relationship between the single fermion cell and single fermion particle and the single boson cell and more than one boson particle in the assumed processes for developing the theory of adsorption, simply by maximising the entropy of the system for both types of particles. In this work, the reduction of the general expression of adsorption to the special case of the Langmuir adsorption isotherm and closely related family of curves or types of adsorption isotherms in dimensionless form have been derived using statistical mechanics for an adsorbate. The classification of the laboratory data, for adsorption distribution concept, power of generalised method in terms of nonlinear parameter least square fits by selecting different sets of derived functional form one by one is demonstrated.