The solution set for minimizing the function $\mathrm{z}=x+\mathrm{y}$ with constraints $x+\mathrm{y} \geqslant 2, x+2 \mathrm{y} \leqslant 8, \mathrm{y} \leqslant 3, x, \mathrm{y} \geqslant 0$ contains

Linear programming is used to minimize the function

$z = x + y$

subject to the constraints:

$x + y \geq 2$, $x + 2y \leq 8$, $y \leq 3$, $x \geq 0$, and $y \geq 0$.

The feasible region is a convex polygon whose corner points are $(2,0)$, $(8,0)$, $(2,3)$, $(0,3)$, and $(0,2)$.

Evaluating the objective at each corner gives
$z(2,0)=2$, $z(8,0)=8$, $z(2,3)=5$, $z(0,3)=3$, and $z(0,2)=2$.

The minimum value is $2$, achieved at both $(2,0)$ and $(0,2)$. Since these are adjacent vertices, all points on the line segment connecting them—where $x + y = 2$ with $0 \leq x \leq 2$ and $0 \leq y \leq 2$—yield the same minimum. There are infinitely many such points.

Final answer: $\boxed{\text{C}}$