  We extend the classical coupon collector's problem to one in which two
collectors are simultaneously and independently seeking collections of $d$
coupons. We find, in finite terms, the probability that the two collectors
finish at the same trial, and we find, using the methods of Gessel-Viennot, the
probability that the game has the following ``ballot-like'' character: the two
collectors are tied with each other for some initial number of steps, and after
that the player who first gains the lead remains ahead throughout the game. As
a by-product we obtain the evaluation in finite terms of certain infinite
series whose coefficients are powers and products of Stirling numbers of the
second kind.
  We study the variant of the original coupon collector's problem in which a
single collector wants to obtain at least $h$ copies of each coupon. Here we
give a simpler derivation of results of Newman and Shepp, and extend those
results. Finally we obtain the distribution of the number of coupons that have
been obtained exactly once (``singletons'') at the conclusion of a successful
coupon collecting sequence.
