187. System with three unknowns. Solve the following system of equations:

$$
\left\{\begin{aligned}
x+y+z & =6 \\
x y+y z+z x & =11 \\
x y z & =6
\end{aligned}\right.
$$
187. Recalling the connection between the roots and coefficients of a third-order equation with one unknown, we notice that $x, y, z$ coincide with the roots of the cubic equation.

$$
a^{3}-6 a^{2}+11 a-6=0
$$

Further,

$$
\begin{gathered}
(a-1)(a-2)(a-3)=0 \\
a=1,2,3
\end{gathered}
$$

Since the original system is symmetric with respect to $x, y, z$, all its six solutions coincide with the six permutations of the numbers 1, 2, 3, namely:

$$
\begin{aligned}
& (x, y, z)=(1,2,3),(1,3,2),(2,1,3) \\
& \quad(2,3,1),(3,1,2),(3,2,1) . \\
& {[\text { D. G. Ballew, S. S. M., 40, } 483 \text { (May 1940).] }}
\end{aligned}
$$