Sulfate aerosols' cooling effect on the global climate has spurred research to understand their mechanisms of formation. Both theoretical and laboratory studies have shown that the formation of sulfate aerosols is enhanced by the presence of a base like ammonia. Stronger alkylamine bases such as monomethylamine (MMA), dimethylamine (DMA), and trirnethylamine (TMA) further increase aerosol formation rates by many orders of magnitude relative to that of ammonia. However, recent lab measurements have found that the presence of ammonia and alkylamines together increases nucleation rates by another 1-2 orders of magnitude relative to the stronger alkylamines alone. This work explores that observation by studying the thermodynamic stability of clusters containing up to two sulfuric acids and two bases of the same or different type. Initial configurational sampling is performed using genetic algorithm (GA) interfaced to semiempirical methods to find a large number of low-energy configurations. These structures are then subject to quantum mechanical calculations using PW91, M06-2X, and omega 97X-D functionals and MP2 with large basis sets. The thermodynamics of formation is reviewed to determine if it rationalizes why mixed base systems yield higher rates of aerosol formation than single base ones. The gas phase basicity of the bases in a cluster is the main determinant of binding strength in smaller clusters such as those in the current study while aqueous phase basicity is more important for larger particles. Besides thermodynamic considerations, the differences in aerosol formation mechanisms as a function of size and between the gas and particle phases are discussed.