Pair collisions are the main interaction process in the Boltzmann gas dynamics. By making use of exactly the same physical assumptions as was done by Ludwig Boltzmann we wrote the kinetic equation for two-particle distribution function of molecules in gas mixtures. Instead of the collision integral, there are the linear scattering operator and the chaos projector in the right part of this equation. We developed a new technique for factorization of the scattering operator on the bases of right inverses to the Casimir operator of the group of rotations. We exactly transformed the Boltzmann collision integral to the Landau-Fokker-Planck like form.