<s>[INST] If $x$, $y$ and $z$ are distinct positive integers and $x+y+z=11$ then what is the maximum value of $(xyz+xy+yz+zx)$? 
If $x$, $y$ and $z$ are distinct positive integers and $x+y+z=11$ then what is the maximum value of $(xyz+xy+yz+zx)$?

We know that product is maximum when difference between $x$, $y$ and $z$ is minimum.
So, I assumed $x=3$, $y=4$ and $z=4$. 
Now putting this value in $xyz+xy+yz+zx$ I got my answer $88$. But actual answer is $78$. Where am I doing it wrong?
 [/INST] Note that $(x+1)(y+1)(z+1)=1+(x+y+z)+(xy+yx+zx)+xyz$ so the sum you want is $$(x+1)(y+1)(z+1)-12$$ which justifies your comment about the product. 
If $a\gt a-1\ge b+1\gt b$ we have $$(a-1)(b+1)=ab+(a-b)-1\gt ab$$ since $a-b\ge 2$. 
The best selection of distinct integers is (as others have noted) $5+4+2=11$. And the product formula gives $90-12=78$.
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