Suppose c is the maximum of a, b, and c; it can be (n-1) tops, so the right hand side is, maximum, 3(n-1)!. If n is greater than 3, that will be less than n!, so n can be 3, tops.
For n=3, a=b=c=2 is the obvious solution.
For n=1 or 2, the left hand side is less than 3, so that cannot work.
One and only one
