ABX(3)-type organic lead halide perovskites currently attract broad attention as light harvesters for solar cells due to their high power conversion efficiency (PCE). Mixtures of formamidinium (FA) with methylammonium (MA) as A-cations show currently the best performance. Apart from offering better solar light harvesting in the near IR the addition of methylammonium stabilizes the perovskite phase of FAPbI(3) which in pure form at room temperature converts to the yellow photovoltaically inactive delta-phase. We observe a similar phenomenon upon adding Cs+ cations to FAPbI(3). CsPbI3 and FAPbI3 both form the undesirable yellow phase under ambient condition while the mixture forms the desired black pervoskite. Solar cells employing the composition Cs(0.2)FA(0.8)PbI(2.84)Br(0.16) yield high average PCEs of over 17% exhibiting negligible hysteresis and excellent long term stability in ambient air. We elucidate here this remarkable behavior using first principle computations. These show that the remarkable stabilization of the perovskite phase by mixing the A-cations stems from entropic gains and the small internal energy input required for the formation of their solid solution. By contrast, the energy of formation of the delta-phase containing mixed cations is too large to be compensated by this configurational entropy increase. Our calculations reveal for the first time the optoelectronic properties of such mixed A-cation perovskites and the underlying reasons for their excellent performance and high stability.