Question

Find the number of natural number solutions to \(a + b + c = 8\)

• A. 24
• B. 28
• C. 30
• D. none of these

Hint:

Whole Numbers (\( \ge 0 \)): \( ^{n+r-1}C_{r-1} \)
Natural Numbers (\( \ge 1 \)): \( ^{n-1}C_{r-1} \)
The natural number count is always smaller because you are "forced" to put at least one item in each box.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Natural number solutions mean $a, b, c \ge 1$. We give 1 to each variable first, then distribute the remaining sum.

Step 2: Key Formula or Approach:
The number of positive integer (natural number) solutions to \( x_1 + x_2 + \dots + x_r = n \) is: \[ ^{n-1}C_{r-1} \]

Step 3: Detailed Explanation:
1. Here, $n = 8$ and $r = 3$.
2. Apply the formula: \( ^{8-1}C_{3-1} = \, ^7C_2 \).
3. Calculation: \[ ^7C_2 = \frac{7 \times 6}{2 \times 1} = 21. \] 4. Since 21 is not listed in the provided options A, B, or C, the correct choice is "none of these."

Step 4: Final Answer:
The number of natural number solutions is 21.