Use Viet system we immediately see that x,y,z are the solutions of the expression :

\(t^3-10t^2+31t-30=0\Leftrightarrow\left[{}\begin{matrix}t=5\\t=2\\t=3\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=5;y=2z=3\\x=5;y=3;z=2\\x=2;y=5;z=3\\x=2;y=3;z=5\end{matrix}\right.\)and \(\left[{}\begin{matrix}x=3;y=2;z=5\\x=3;y=5;z=2\end{matrix}\right.\)

So we have six pairs of solutions : \(\left(5;2;3\right),\left(5;3;2\right),\left(2;5;3\right),\left(2;3;5\right),\left(3;2;5\right),\left(3;5;2\right)\)