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In this paper, we examine the problem of count data clustering. We analyze this problem using finite mixtures of distributions. The multinomial distribution and the multinomial Dirichlet distribution (MDD) are widely accepted to model count data. We show that these two distributions cannot be the best choice in all the applications, and we propose another model called the multinomial generalized Dirichlet distribution (MGDD) that is the composition of the generalized Dirichlet distribution and the multinomial, in the same way that the MDD is the composition of the Dirichlet and the multinomial. The estimation of the parameters and the determination of the number of components in our model are based on the deterministic annealing expectation-maximization (DAEM) approach and the minimum description length (MDL) criterion, respectively. We compare our method to standard approaches such as multinomial and multinomial Dirichlet mixtures to show its merits. The comparison involves different applications such as spatial color image databases indexing, handwritten digit recognition, and text document clustering.