Let $(X_1,\dots, X_M) \sim Mult(N;p_1,\dots,p_M)$ 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 $n\in \{0,\dots,N\}$ is fixed?
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