On the asymptotic behaviour of the probability of existence of equivalent tuples with nontrivial structure in a random sequence

In a long enough sequence of discrete random variables, as a rule, an $s$-tuple exists of nontrivial structure, that is, a tuple with at least one repeated symbol. We consider the case where the sequence consists of $n+s-1$ independent random variables taking the values $1,\dots,N$ with equal probabilities. It is shown that as $n\to\infty$, $ns^3N^{-2}\to0$ the probability of that in the sequence $s$-tuples exist with the same nontrivial structure is equal to $1-(1+n/N)^se^{-sn/N}(1+o(1))$.