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Starting from a neglected work by Rappeport (1968), we re-propose an exact algorithm to compute the distribution of the maximum of a multinomial random vector under the hypothesis of equiprobability. We then show how to compute the exact probabilities of the sum of the J largest order statistics of the vector, following the suggestions and correcting the errors in the same article. Finally, we introduce brand new ways of computing the exact probabilities of the multinomial minimum and of the multinomial range. The exact probabilities we derive can be used in all those situations in which the multinomial distribution plays an important role, from goodness-of-fit tests to the study of Poisson processes, with applications spanning from biostatistics to finance. For all algorithms, we provide Matlab codes and ready-to-use tables of critical values.