A Comparison Of Two Methods For Random Labelling of Balls by Vectors of Integers

Kirk (1993) raised the question of comparing the following two ways for labelling balls. Given r pre-determined positive integers n _i (1 ≤ i ≤ r), and given N balls (N large), consider two ways to randomly assign r–component vectors of integers (a₁,…, a _r ) to them, such that 1 ≤ a _i ≤ n _i . We will call these vectors labels. Of course, altogether there are $$\prod {_{i = 1}^r \,n_i }$$ possible labels.