Given that,
$$
\begin{gathered}
x+y+z=4, \quad x-y+z=2 \text { and } x+2 y+2 z=1 \\
\left.\Delta=\left|\begin{array}{ccc}
1 & 1 & 1 \\
1 & -1 & 1 \\
1 & 2 & 2
\end{array}\right|=1(-2-2)-1(2-1)+(2+1)\right] \\
\Delta_1=\left|\begin{array}{ccc}
4 & 1 & 1 \\
2 & -1 & 1 \\
1 & 2 & 2
\end{array}\right|=4(-2-2)-1(4-1)+1(4+1) \\
=4(-4)-3+5=-16-3+5=-14 \\
\Delta_2=\left|\begin{array}{ccc}
1 & 4 & 1 \\
1 & 2 & 1 \\
1 & 1 & 2
\end{array}\right| \\
\Delta_2=1\left(\begin{array}{lll}
(4-1)-4(2-1)+1(1-2)=3-4-1=-2
\end{array}\right. \\
\Delta_3=\left|\begin{array}{ccc}
1 & 1 & 4 \\
1 & -1 & 2 \\
1 & 2 & 1
\end{array}\right| \quad y=\frac{\Delta_2}{\Delta}=\frac{-2}{-2}=1 \\
\Delta_3=1(-1-4)-1(1-2)+4(2+1)=(-5)+1+12=8 \\
x=\frac{\Delta_1}{\Delta}=\frac{-14}{-2}=7, \quad \begin{array}{l}
8 \\
\text { and } z=\frac{\Delta_3}{\Delta}=\frac{8}{-2}=-4 \quad y
\end{array}
\end{gathered}
$$


So, $a=7, b=1, c=-4$
Now, $a b+b c+c a$
$$
=7(1)+1(-4)+(-4)(7)=7-4-28=-25
$$