Proof of intractability of computing the prior distribution in bayesian inference

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In this paper: https://www.tandfonline.com/doi/epdf/10.1080/01621459.2017.1285773?needAccess=true I do not understand why there are K^n terms in the integral in Equation (9) instead of n^K. If we have n observations, and each observation can belong to one of K clusters, then we should have n^K possible assignments. Can anyone clarify?