We consider the semigroup generated by random mappings and random bijective mappings of a finite set Ω_n of cardinality n into itself. We study the question when this semigroup includes all mappings of Ω_n into itself with a fixed cardinality k of the image of the set Ω_n. As n → ∞, the ranges of k are given where this inclusion holds with probability tending to zero or one, and two domains of values of k where the inclusion holds with intermediate probability.
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