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This is the ordinary factorial of $X$, rather than a falling-factorial moment. Substituting the probability mass function of the Poisson distribution gives the nonnegative seriesThe geometric series converges exactly when $0<\lambda <1$. Therefore $\mathbb{E}[X!]=\frac{e^{-\lambda }}{1-\lambda }$ for $0<\lambda <1$. For $\lambda \ge 1$, its terms fail to tend to zero and the expectation is infinite. This is the factorial of a Poisson random variable.