If x, y, z are positive integers then value of the expression (x + y) (y + z) (z + x) is

Let x, y, z be positive integers then (x + y) (y + z) (z + x) = ??

Since $\frac{x+y}{2}>\sqrt{x y}$ (Since arithmetic mean $\geq$ geometric mean)

i. e $x+y>2 \sqrt{x y}$

Similarly $y+z>2 \sqrt{y z}$ and $z+x>2 \sqrt{z x}$

$\therefore(x+y)(y+z)(z+x)>8 \sqrt{x y} \sqrt{y z} \sqrt{z x}$

$=8 \sqrt{x^{2} y^{2} z^{2}}$

$=8 x y z$

$\therefore(x+y)(y+z)(z+x)>8 x y z$

Hence, the correct answer is option B.