<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] $11$ is reasonably small. So we can search exhaustively through the options to find maximum $f=xyz+xy+yz+zx$.
Take $x<y<z$. Then smallest values for $x,y$ are $1,2$ and thus greatest $z=8$. Then we can work down values of $z$ and partition the residue accordingly, avoiding duplicates.
$\begin{array}{|c|c|} \hline
(x,y,z) & f\\ \hline
(1,2,8) & 42 \\
(1,3,7) & 52 \\
(1,4,6) & 58 \\
(2,3,6) & 72 \\
(2,4,5) & 78 \\ \hline
\end{array}$
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