Why can't the factorion of n digit number exceed n*9!

Question

"A factorion is an integer which is equal to the sum of factorials of its digits."

I read from mathworld that the factorion of a $n$-digit number cannot exceed $n\cdot 9!$.

Why is this? I mean what is the proof?

Answer 1

The largest an $n$-digit number can be is a concatenation of $n$ nines: $$\underbrace{9\,9\,\cdots 9}_{n\text{-digits long}}$$

The associated factorion to this largest possible $n$-digit number is $$\underbrace{9!+ 9! + \cdots + 9!}_{n \;\text{ summands of }\; 9!} = n \cdot 9!$$

Answer 2

If a number is a factorion, then it is equal to the sum of the factorials of its digits. It only has n digits and each digit is less than or equal to 9, hence the factorian is less than or equal to $ 9! n $.