# Challenge 396: Factorial Sum

Investigate which numbers have this property involving factorial sums.

For this challenge you need to know that the notation "!" after a positive integer n means the product of all the integers up to and including n. So, for example,

4! = 4 x 3 x 2 x 1 = 24

and 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3,628,800

Here's the problem!

(i) Are there any single digit numbers n for which n! = n? If so what are they? If not, prove it!

(ii) Are there any two digit numbers ab (meaning10a + b) for which ab = a! + b! If so what are they? If not, prove it!

[Example: ab cannot be 23, because 2! + 3! is 8, not 23]

(iii) Are there any three digit numbers abc (meaning100a + 10b + c) for which abc = a! + b! + c! If so what are they? If not, prove it!

[Example: abc cannot be 235, because 2! + 3! + 5! is 128, not 235]