  This paper provides analysis to a generalized version of the coupon collector
problem, in which the collector gets $d$ distinct coupons each run and she
chooses the one that she has the least so far. On the asymptotic case when the
number of coupons $n$ goes to infinity, we show that on average $\frac{n\log
n}{d} + \frac{n}{d}(m-1)\log\log{n}+O(mn)$ runs are needed to collect $m$ sets
of coupons. An efficient exact algorithm is also developed for any finite case
to compute the average needed runs exactly. Numerical examples are provided to
verify our theoretical predictions.
