Understanding the Problem
To find the number of positive integral solutions to the inequality 3x + y + z ≥ 30, we first rewrite the inequality in a more manageable form.
Rearranging the Inequality
We can express this as:
3x + y + z = k, where k ≥ 30.
This means we need to find the number of solutions for each integer k starting from 30.
Finding Positive Solutions
Given that x, y, and z must be positive integers:
- The minimum value for y and z is 1.
- Thus, we can rewrite the equation as:
3x + (y-1) + (z-1) = k - 2.
Letting a = y - 1 and b = z - 1, we have:
3x + a + b = k - 2, with a, b ≥ 0.
Using Stars and Bars Method
Now, let m = k - 2:
3x + a + b = m.
To find the number of solutions for fixed x:
- Rewrite it as a + b = m - 3x, ensuring that m - 3x ≥ 0 (thus x ≤ m/3).
- The number of non-negative solutions for a + b = n is given by (n + 1) choose 1.
Calculating Solutions for Each k
For each value of k starting from 30:
1. Calculate m = k - 2.
2. Count the number of x values (1 to m/3).
3. For each x, calculate the corresponding non-negative solutions for a and b.
Finally, sum the solutions across all valid k values to get the total number of positive integral solutions.
This method ensures all combinations are counted, providing a comprehensive solution to the inequality.