If the difference between the maximum and minimum values of the objective function $\mathrm{z}=7 x-8 \mathrm{y}$ subject to the constraints $x+y \leqslant 20$, $y \geqslant 5, x, y \geqslant 0$ is $5 k+200$, then the value of $k$ is

Identifying feasible region vertices

The feasible region is bounded by the constraints: $x + y \leq 20$, $y \geq 5$, and $x \geq 0$.
Intersection of $x = 0$ and $y = 5$ gives vertex $(0,5)$.
Intersection of $y = 5$ and $x + y = 20$ gives vertex $(15,5)$.
Intersection of $x = 0$ and $x + y = 20$ gives vertex $(0,20)$.

Evaluating objective function

$z = 7x - 8y$ at $(0,5)$ yields $z = -40$, at $(15,5)$ yields $z = 65$, and at $(0,20)$ yields $z = -160$.
The maximum is $65$ and the minimum is $-160$.

Solving for k

$\text{Maximum} - \text{Minimum} = 65 - (-160) = 225$

$225 = 5k + 200$

$5k = 25$

$k = 5$

The value is $\boxed{C}$.