On the numbers of equivalent tuples sets in a sequence of independent random variables

Let $\mathbf X$ be a sequence of $n+s-1$ polynomial trials with $N$ outcomes. Limit joint distributions of the numbers of $r$-sets of equivalent $s$-tuples in $\mathbf X$ are proved. Two types of conditions on the parameters $n,N\to\infty$, $s<N$ are considered. Under the conditions of the first type the mean number of $s$-tuples with coinciding outcomes is bounded. Under the conditions of the second type the mean number of $s$-tuples without concidings is bounded.