Do \(3\le a;b;c\le5\Rightarrow\left\{{}\begin{matrix}\left(a-3\right)\left(b-3\right)\left(c-3\right)\ge0\\\left(5-a\right)\left(5-b\right)\left(5-c\right)\ge0\end{matrix}\right.\)
\(\Rightarrow\left(a-3\right)\left(b-3\right)\left(c-3\right)+\left(5-a\right)\left(5-b\right)\left(5-c\right)\ge0\)
\(\Leftrightarrow2\left(ab+bc+ca\right)-16\left(a+b+c\right)+98\ge0\)
\(\Leftrightarrow a^2+b^2+c^2+2\left(ab+bc+ca\right)-16\left(a+b+c\right)+48\ge0\)
\(\Leftrightarrow\left(a+b+c\right)^2-16\left(a+b+c\right)+48\ge0\)
\(\Leftrightarrow\left(a+b+c-4\right)\left(a+b+c-12\right)\ge0\)
Mặt khác \(a;b;c\ge3\Rightarrow a+b+c\ge9>4\Rightarrow a+b+c-4>0\)
\(\Rightarrow a+b+c-12\ge0\)
\(\Rightarrow a+b+c\ge12\)
\(P_{min}=12\) khi \(\left(a;b;c\right)=\left(3;4;5\right)\) và các hoán vị
