Answer: D

Small Hint:

Group solutions according to $s=x+y$

Big Hint:

For fixed $s,$ count the positive ordered pairs $(x,y)$ and require $n-2s\ge1$

Solution:

For $s=x+y\ge2,$ there are $s-1$ positive ordered pairs $(x,y),$ and $z=n-2s$ is positive when $s\le\lfloor\frac{n-1}{2}\rfloor.$ Writing $m=\lfloor\frac{n-1}{2}\rfloor,$ the number of solutions is $$\sum_{s=2}^{m}(s-1)=\frac{m(m-1)}2.$$ Setting this equal to $28$ gives $m=8.$ Hence $\lfloor\frac{n-1}{2}\rfloor=8,$ so $n=17$ or $18.$

Thus the correct answer is D.