<p>Find the number of ways of selecting 10 balls out of an unlimited number of identical white, red, and blue balls.</p>
Step-by-Step Solution
Key Concept: This is a 'distributions' problem requiring stars-and-bars method: distributing 10 identical balls among 3 distinct colors is equivalent to finding non-negative integer solutions to w + r + b = 10, which equals C(10+3-1, 3-1) = C(12, 2).
<p><strong>Step 1:</strong> Identify the problem type. We need to select 10 balls from 3 types (colors) where balls of the same color are identical. This is a 'distribution of identical objects' problem.</p><p><strong>Step 2:</strong> Let w, r, b represent the number of white, red, and blue balls selected respectively. We need: w + r + b = 10, where w, r, b ≥ 0 (non-negative integers).</p><p><strong>Step 3:</strong> Apply stars-and-bars formula. The number of ways to distribute n identical objects into k distinct groups is C(n+k-1, k-1).</p><p><strong>Step 4:</strong> Here n = 10 (balls) and k = 3 (colors), so the answer is: C(10+3-1, 3-1) = C(12, 2) = 12×11/2 = 66.</p><p>∴ Answer: <strong>{}^{12}C_2</strong></p>
Correct Answer: \({}^{12}C_2\)