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We consider the polynomial scheme of trials with outcomes E 0 , E 1 , ... , E N  and the corresponding probabilities p 0 , p 1  , ... , p N . We assume that the trials are performed until the rth occurrence of the outcome E 0 , r = 1, 2, ... If η j (r) is the number of occurrences of the outcome E j  at the stopping time, j = 1, ... , N, and η(r) = (η 1 (r), ... , η N (r)), then the vector η(r) has the negative polynomial distribution. Under the assumptions that N ∈ ℕ and the positive probabilities p 0 , p 1 , ... , p N  are fixed, that r → ∞ and k 1 , ... , k N  → ∞ so that the parameters .β j  = k j /r satisfy the inequalities β j  ≥ ε, where ε is a positive constant, j = 1, ... , N, and under some additional constraints, we give asymptotic estimates of the probabilities of large deviations

On the asymptotic behaviour of probabilities of large deviations for the negative polynomial distribution