Balls are drawn one after another fromkcellsC1, …,Ckaccording to the multinomial distribution. Suppose theith cell initially containsNiballs, and sampling stops as soon as any of thekcells, sayCα, empties first. LetXidenote the number of balls taken from cellCi(alli ≠ α) at stopping time. The joint asymptotic (asNi→ ∞) distribution of theXi(i ≠ α) is derived under the most general configuration of the multinomial cell probabilitiesp1, …,pk.Conditions onpiandNiare given under which the asymptotic distribution is shown to be either normal or truncated (restricted) normal. An application of the asymptotic distribution theory forNi=N0(i= 1, …,k) in setting up approximate tests and confidence intervals for the largestpiis also given. Under certain conditions onpiandNiit is shown that the asymptotic probability thatCiempties first is equal to the probability content of a positive orthant under a multivariate normal distribution.