Product $=$ Sum

We want to find the largest 3 digit number $\overline{abc}$ such that the product of its digits is equal to the sum of its digits. WLOG, $a \geq b \geq c$ . $a+b+c=abc$ $\Rightarrow abc \leq 3a$ $\Rightarrow bc \leq 3$ $\Rightarrow c=1,0$

Case 1: $c=1$ $\Rightarrow a+b+1=ab$ $\Rightarrow (a-1)(b-1)=2$ $\Rightarrow (a,b)=(3,2)$ Thus, $(3,2,1)$ is a solution for $(a,b,c)$

Case 2: $c=0$ $\Rightarrow a+b=0$ $\Rightarrow (a,b)=(0,0)$

Thus, $(0,0,0)$ is also a solution.

We can remove the second solution since it doesn't have a permutation which makes a 3 digit number. The arrest value of the permutations made by the first solution is 321. Therefore, the answer is 321.