This is a nightmare of a question if one does by cases.Find the number of integral solutions of

x1 + x2 + x3 + x4 = 28

where each xi ≥ 1, and in addition each xi ≤ 8. Write your answer in terms of binomial coefficients.

It is easy if you use generating functions.

Expand the answer to this question is the coefficient to the term.

Luckily the is an online resource to do just that.