The general collector’s problem describes a process in which N distinct coupons are placed in an urn and randomly selected one at a time (with replacement) until at least m of all N existing diﬀerent types of coupons have been selected. Let Tm(N) the random variable denoting the number of trials needed for this goal. We brieﬂy present the leading asymptotics of the (rising) moments of Tm(N) as N → ∞ for large classes of coupon probabilities. It is proved that the expectation of Tm(N) becomes minimum when the coupons are uniformly distributed. Moreover, a theorem on the asymptotic estimates of the rising moments of Tm(N) by comparison with known sequences of coupon probabilities is proved.