For each of the following, determine a generating function and indicate the coefficient in the function that is needed to solve the problem. (Give both the polynomial and power series forms of the generating function, wherever appropriate.) Find the number of integer solutions for the following equations
(a) c1 + c2 + C3 + C4 = 20, 0 < ct < 7 for all 1 < i < 4
(b) c1 + c2 + C3 + C4 = 20, 0 < ct for all 1 < i < 4, with c2 and C3 even
(c) c1 + c2 + C3 + C4 + c5 = 30, 2 < ci < 4 and 3 < ct < 8 for all 2 < i < 5
(d) ci + c2 + C3 + C4 + c5 = 30, 0