I'm working on a problem where I need to count the number of integer solutions for the inequality $10 \leq x_1 + x_2 + x_3 + x_4 \leq 19$, given that each $x_i$ (for $i = 1$ to $4$) is bounded by $-5 \leq x_i \leq 10$. I approached the problem in two main steps, transforming the inequalities to work with non-negative integers by adjusting the bounds and then applying combinatorial reasoning. However, I'm unsure if my methodology and calculations are correct. Here's my approach:
Step 1: Dealing with the upper bound First, I transformed the original inequality $x_1 + x_2 + x_3 + x_4 \leq 19$ to a new form where I could deal with non-negative integers by introducing a new variable $B$ and adjusting the bounds of $x_i$:
$x_1 + x_2 + x_3 + x_4 + B = 19$, with $-5 \leq x_i \leq 10$ Then, I adjusted the variables to work within a non-negative range, leading to: $x_1 + x_2 + x_3 + x_4 + B = 39$, with $0 \leq x_i \leq 15$, $B \geq 0$
From there, I calculated the total number of solutions without restrictions, subtracted the cases where one variable exceeds its bound, and added the cases where two variables exceed their bounds. The solution for step 1 was given by: $CR(5,39) - (CR(5, 39-16) \times 4) + (CR(5, 39-32) \times C(4,2))$
Step 2: Dealing with the lower bound Similarly, for the lower bound $10 \leq x_1 + x_2 + x_3 + x_4$, I considered cases where the sum is less than or equal to 9 (to subtract them from the total count): $x_1 + x_2 + x_3 + x_4 + B = 9$, with $-5 \leq x_i \leq 10$ This was transformed to: $x_1 + x_2 + x_3 + x_4 + B = 29$, with $0 \leq x_i \leq 15$, $B \geq 0$
I then calculated the total number of solutions without restrictions for this case and subtracted the cases where one variable exceeds its bound. There were no cases for exceeding two variables. The solution for step 2 was: $CR(5, 29) - (CR(5, 29-16) \times 4)$
Final Solution: Combining the solutions from both steps, I derived the complete solution for the exercise as: $CR(5,39) - (CR(5, 39-16) \times 4) + (CR(5, 39-32) \times C(4,2)) - (CR(5, 29) - (CR(5, 29-16) \times 4))$
I'm seeking feedback on whether my approach and the steps I've taken are correct for solving this problem, particularly my use of combinatorial reasoning and the transformation of inequalities. Any insights into errors in my methodology or calculations would be greatly appreciated.