Show that the equation n! = a_1!a_2!… a_k!, with n-1>a_1≥ a_2≥ … ≥ a_k≥ 2, has only finitely many solutions.

Erdős Problem #373 — Show that the equation $n! = a_1!a_2!\cdots a_k!,$ with $n-1>a_1\geq a_2\geq \cdots \geq a_k\geq 2$ , has only finitely many solutions.

Show that the equation $n! = a_1!a_2!\cdots a_k!,$ with $n-1>a_1\geq a_2\geq \cdots \geq a_k\geq 2$ , has only finitely many solutions.

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