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In the collector’s problem with group drawings, s out of n different types of coupon are sampled with replacement. In the uniform case, each s-subset of the types has the same probability of being sampled. For this case, we derive a Poisson limit theorem for the number of types that are sampled at most 
$c-1$
times, where 
$c \ge 1$
is fixed. In a specified approximate nonuniform setting, we prove a Poisson limit theorem for the special case 
$c=1$
. As corollaries, we obtain limit distributions for the waiting time for c complete series of types in the uniform case and a single complete series in the approximate nonuniform case.
