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Let $(X_1,\ldots,X_M)\sim \operatorname{Mult}(N;p_1,\ldots,p_M)$ follow a multinomial distribution. What is the probability that at least one of the variables takes a certain value, i.e. $\mathbb{P}(\exists k\in \{1,\dots,M \}:X_k=n)$, where $n\in \{0,\ldots,N\}$ is fixed?

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For any nonempty set $S = \{s_1,\ldots, s_k\} \subseteq \{1,\ldots,M\}$ with $nk \le N$, let $$P_S = P(\text{ all } X_{s_j} = n) = \frac{N!}{(n!)^k(N-nk)!} p_{s_1}^n \ldots p_{s_k}^n (1-p_{s_1}-\ldots - p_{s_k})^{N-nk}$$ By inclusion-exclusion, your probability is $\sum_S (-1)^{|S|-1} P_S$, where the sum is over all nonempty $S$ with $n|S|\le N$.

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