Answer: E

Small Hint:

Rescale the variables by setting $x=2u,$ $y=v,$ and $z=\frac{w}{2}$

Big Hint:

The new variables have elementary symmetric sums $6,$ $11,$ and $6$

Solution:

Set $x=2u,$ $y=v,$ and $z=\frac{w}{2}.$ Dividing the first two equations by $2$ gives $$\begin{aligned} u+v+w&=6,\\ uv+vw+uw&=11, \end{aligned}$$ while $uvw=6.$ Thus $u,$ $v,$ and $w$ are the three roots of $p(t)=t^3-6t^2+11t-6.$ This polynomial factors as $$p(t)=(t-1)(t-2)(t-3),$$ so they are $1,$ $2,$ and $3$ in any order. Their $3!=6$ permutations produce $6$ distinct ordered triples $(x,y,z).$

Therefore, the correct answer is E.