In the following question, we will count distinct integer solutions to the equation x1 + x2 + x3 + x4 = 35. (a) how many solutions are there if all the variables must be non-negative? (b) how many solutions are there if all the variables must be positive? (c) how many solutions are there if x1 ≥ 2, x2 ≥ 4, x3 ≥ 0, and x4 ≥ 0? (d) how many solutions are there to the equation x1 + x2 + x3 + x4 ≤ 35 in which all the xi are non-negative? (hint: add an extra variable y such that y is non-negative and x1 + x2 + x3 + x4 + y = 35.)