Minimum value of the expression given below. Is $a,b,c$ are three different positive integers such that :
$$
ab+bc+ca\geq 107
$$
Then what is the minimum value of $a^3+b^3+c^3-3abc$.
I expanded this expression to $(a+b+c)((a+b+c)^2-3(ab+bc+ca))$ and tried to find the minimum value of $(a+b+c)$ by AM-GM inequalities but the problem is that the minimum value occurs at scenario where $a=b=c$, which is ommitted by assumptions of this question.
I'd appreciate some help.

Yes thats the correct answer. We can do it as $3(ab+bc+ca)>321$
and for the value to be non negative (a+b+c)^2 should be just greater than 321 so we can suppose it to be 18 as 18^2=324 and hence (a+b+c)>18
for minimum value it should 18.
and also abc should be greatest for minimum value
so we struck to values 5,6,7 . 
and the minimun value will be 54
