First, we characterize the notion of total variation cutoff for absorbing Markov chains in terms of their absorption times. Then, we consider a problem introduced by Paul Erdős and Alfréd Rényi concerning the distribution of the time required for a coupon collector to obtain m complete sets of n distinct coupons, under the assumption that duplicate coupons cannot be exchanged. For this problem, we investigate an approach to the regime in which m grows with n, a challenging case that has remained open since 1961.