The magnitude of the gradient of a function measures the rate of change of the function in all directions at a specific point. In this case, we are given the function f = xyz^3 and we need to find the magnitude of its gradient at the point (1,0,2).

To find the gradient of the function, we need to take the partial derivatives of the function with respect to each variable (x, y, and z).

The partial derivative of f with respect to x, denoted as ∂f/∂x, can be found by treating y and z as constants and differentiating only the term involving x. In this case, the derivative of xyz^3 with respect to x is yz^3.

Similarly, the partial derivative of f with respect to y, denoted as ∂f/∂y, can be found by treating x and z as constants and differentiating only the term involving y. In this case, the derivative of xyz^3 with respect to y is xz^3.

Finally, the partial derivative of f with respect to z, denoted as ∂f/∂z, can be found by treating x and y as constants and differentiating only the term involving z. In this case, the derivative of xyz^3 with respect to z is 3xyz^2.

Now that we have the partial derivatives, we can find the gradient of the function by combining these derivatives into a vector. The gradient vector is given by ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z).

In this case, the gradient vector is (∂f/∂x, ∂f/∂y, ∂f/∂z) = (yz^3, xz^3, 3xyz^2).

To find the magnitude of the gradient, we calculate the Euclidean norm of the gradient vector, which is given by ||∇f|| = sqrt((∂f/∂x)^2 + (∂f/∂y)^2 + (∂f/∂z)^2).

Substituting the partial derivatives into the formula, we get ||∇f|| = sqrt((yz^3)^2 + (xz^3)^2 + (3xyz^2)^2).

Simplifying the expression, we have ||∇f|| = sqrt(y^2z^6 + x^2z^6 + 9x^2y^2z^4).

Now we substitute the values of x = 1, y = 0, and z = 2 into the expression to find the magnitude of the gradient at the point (1,0,2).

||∇f|| = sqrt(0^2 * 2^6 + 1^2 * 2^6 + 9 * 1^2 * 0^4).

Simplifying further, we have ||∇f|| = sqrt(0 + 64 + 0) = sqrt(64) = 8.

Therefore, the magnitude of the gradient of the function f = xyz^3 at the point (1,0,2) is 8.

Hence, the correct answer is option 'C'.