Step 1: Understanding the Concept:
This is a distribution problem of identical items into distinct groups with lower bounds. We use the "Stars and Bars" method.
Step 2: Key Formula or Approach:
Formula: \( \binom{n + r - 1}{r - 1} \) where \(n\) is the number of remaining items after satisfying the minimum requirements and \(r\) is the number of groups.
Step 3: Detailed Explanation:
1. Total pens = 8.
2. Minimum requirements:
Aal \(\ge 1\), Bal \(\ge 2\), Cal \(\ge 3\).
3. First, give the minimum number of pens to each:
Pens given = \(1 + 2 + 3 = 6\) pens.
4. Remaining pens to distribute freely = \(8 - 6 = 2\) pens.
5. Distribute \(n=2\) identical pens among \(r=3\) people:
Ways = \( \binom{2 + 3 - 1}{3 - 1} = \binom{4}{2} \).
6. Calculation: \( \binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6 \).
Step 4: Final Answer:
There are 6 ways to distribute the pens.