These will be two separate talks, each one for a half-hour. First I'll discuss a new method, jointly due to Warren Ewens, of the U of P., and myself, for exactly computing the maximum of the multinomial distribution, in linear time, with very small storage, and a very short program. I'll give applications of this to deciding if apparent clusters of childhood leukemia cases in two U.S. cities really are unusually concentrated or not. Second, I'll present a short proof of a refinement of Ramanujan's partition congruence mod 5, which is based on the theory of 2-cores and the so-called "BG-rank" of an integer partition.