OK Mea culpa  perhaps This probability refers to the probability for sets with unique elements only Probably other readers too think of This probability to refer to the probability of the birthday paradox Again thats why I say that the birthday paradox is better understood via the mathematics of sets More deeply there are no formulae but algorithms Thus the cases of absurdity for N  M can be avoided The formula of arrangements to calculate the number of sets with unique elements only leads to an absurdity You cant arrange  dice taken  at a time