Determine the Number of Integer Solutions $x_1 + x_2 + x_3 + x_4 = 32$ with restrictions The Question

My Problem
Part a is straight forward, just $C(35,32)$. I'm having a little difficulty with the restrictions and understanding what they mean. $x_1 > 0$ means we shouldn't have any solutions of the form $(0 + 32)$? If that's the case how would I factor that into my answer? I'm thinking taking the answer to part a and subtracting all the solutions which contain $32+0$ but I don't really know how to count those.

For part f) we can use generating functions. The one that models f is
$$G(x)=\left(x+x^2+x^3+...+\right)^3 \left(x+x^2+...+x^{25}\right)$$
these are useful because they reduce combinatoric problems to computational ones. The above gf can be written:
$$g=\left(\frac{x}{1-x}\right)^3 \cdot \frac{ \left(x-x^{26}\right)}{1-x}$$
Now we just need to get at the coefficient of $x^{32}$. I have done that already for you to keep the post small. The term of interest is $4475x^{32}$. So the answer is 4475.
