The most abundant fullerenes, C(60) and C(70), and all the pure carbon fullerenes larger than C(70), follow the isolated-pentagon rule (IPR). Non-IPR fullerenes containing adjacent pentagons (APs) have been stabilized experimentally in cases where, according to Euler's theorem, it is topologically impossible to isolate all the pentagons from each other. Surprisingly, recent experiments have shown that a few endohedral fullerenes, for which IPR structures are possible, are stabilized in non-IPR cages. We show that, apart from strain, the physical property that governs the relative stabilities of fullerenes is the charge distribution in the cage. This charge distribution is controlled by the number and location of APs and pyrene motifs. We show that, when these motifs are uniformly distributed in the cage and well-separated from one other, stabilization of non-IPR endohedral and exohedral derivatives, as well as pure carbon fullerene anions and cations, is the rule, rather than the exception. This suggests that non-IPR derivatives might be even more common than IPR ones.