4. Invent an interesting positive integer n with 40 n. Consider the problem of counting nonnegative integer solutions to (1) That satisfy the conditions 0 ≤ 10,0 15,0 <3 <7,6< 24- 1 (a) (10 points) How many total nonnegative integer solutions are there to (1) (not just solutions that satisfy the given conditions, but all solutions)? (b) (10 points) Define categories C1, C2, C3, and c, such that the number of integer solutions to (1) that do satisfy the given conditions is N (₁₂). (c) (10 points) What is N(c)? (d) (10 points) What is N(c₁₂)? (e) (10 points) Write down and simplify as much as possible a gener- ating function f(x) such that the coefficient of x in f(x) gives the number of integer solutions to (1) that satisfy the given con- ditions.