On maximal quantity of particles of one color in analogs of multicolor urn schemes

We deal with analogs of multicolor urn schemes such that the number of particles is not more than a given number. We introduce conditions which provide the convergence of random variables which is the maximal number of taken particles of a same color to a random variable that has values zero and one. We prove this convergence in the case when a number of taken particles is not more than a fixed number and number of colors converges to infinity. We also consder the case when the number of taken particles converges to infinity.