On factorial reciprocals in Cantor sets

Let $C$ be the middle-third Cantor set. We show that $$\{\frac{1}{n!}:n\in \mathbb{N}\}\cap C=\{1,\frac{1}{5!}\}\mathrm{.}$$ This answers a question recently posed by Jiang [J. Lond. Math. Soc., 2026, published online]. Our approach generalizes to general missing-digit sets, showing that, in any such set, there are only finitely many elements of the form $\frac{1}{n!}$, all of which can be effectively determined.