Question

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

• A. \(^{8}P_{3}\)
• B. \(^{8}C_{3}\)
• C. \(8^{3}\)
• D. none of these

Hint:

For whole number solutions (\( \ge 0 \)), always use the n + r - 1 formula. Think of it as arranging 8 stars and 2 bars (to create 3 spaces).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This is a "Stars and Bars" problem. Whole number solutions mean $a, b, c \ge 0$. We are distributing 8 identical units into 3 distinct bins.

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

Step 3: Detailed Explanation:
1. Here, $n = 8$ and $r = 3$ (variables $a, b, c$).
2. Apply the formula: \( ^{8+3-1}C_{3-1} = \, ^{10}C_2 \).
3. Calculation: \[ ^{10}C_2 = \frac{10 \times 9}{2 \times 1} = 45. \] 4. Since none of the specific choices (A, B, C) equal 45, the answer is "none of these."

Step 4: Final Answer:
The number of whole number solutions is 45.