$\int_{0} \frac{8-x-y}{2} \int_{0}^{8-x-2 z} \int_{0}^{8-y-2 z}(x y z) d z d y d x$

Step 1 ：Given the triple integral problem: \(\int_{0} \frac{8-x-y}{2} \int_{0}^{8-x-2 z} \int_{0}^{8-y-2 z}(x y z) d z d y d x\)

Step 2 ：The integrand is \(xyz\) and the limits of integration are given for \(x\), \(y\), and \(z\). The limits of \(z\) depend on \(x\) and \(y\), the limits of \(y\) depend on \(x\), and \(x\) ranges from 0 to 8.

Step 3 ：We will solve this problem by first integrating with respect to \(z\), then \(y\), and finally \(x\).

Step 4 ：First, integrate with respect to \(z\): \(x*y*(-y - 2*z + 8)^2/2\)

Step 5 ：Next, integrate with respect to \(y\): \(x*(-x - 2*z + 8)^4/8 + (2*x*z/3 - 8*x/3)*(-x - 2*z + 8)^3 + (-x - 2*z + 8)^2*(x*z^2 - 8*x*z + 16*x)\)

Step 6 ：Finally, integrate with respect to \(x\): \(64*z^4/3 - 1024*z^3/3 + 2048*z^2 - 16384*z/5 + 32768/15\)

Step 7 ：The final integral does not depend on any variables, so it is a constant. This is the final answer.

Step 8 ：Final Answer: \(\boxed{\frac{64}{3}z^4 - \frac{1024}{3}z^3 + 2048z^2 - \frac{16384}{5}z + \frac{32768}{15}}\)