Quota Fulfillment , Random Walks and Matching

Suppose an urn contains n sets of beads, each set of size k. Beads are drawn from the urn one at a time without replacement. We derive a closed-form expression for the expected number of draws required to completely remove the first k-set, the second, the third, etc. We accomplish this by recasting the problem as a special kind of random walk through the integer points in [0, k]n and exploiting the symmetries of the hypercube. We also show that the expected proportion of beads removed when the first k-set is completed is between two constant multiples of k p 1/n.