Abstract: Consider a sequence of multinomial random vectors with increasing number of equiprobable cells. We show that if the number of trials increases fast enough, the sequence of the maximum of successes in each cells converges to Gumbel distribution after suitable centering and scaling. We shall also consider the maximum of the coordinates of Dirichlet distribution of increasing dimension. In the process, we shall also characterize the limit of the maximum of a triangular array of iid Gamma random variables with unit scale and changing shape parameters.