<p>Find number of non-negative integral solutions of the equation \(x + y + z = 10\).</p>
Step-by-Step Solution
Key Concept: Use stars and bars theorem: distributing n identical objects into k bins equals C(n+k-1, k-1). Here, distributing 10 units among 3 variables gives C(10+3-1, 3-1) = C(12, 2).
<p><strong>Step 1:</strong> Identify the problem type. We need non-negative integer solutions to x + y + z = 10, which is a 'distribution of identical objects' problem.</p><p><strong>Step 2:</strong> Apply stars and bars formula. We have n = 10 (sum value) and k = 3 (number of variables). The formula for non-negative solutions is C(n+k-1, k-1).</p><p><strong>Step 3:</strong> Calculate: C(10+3-1, 3-1) = C(12, 2) = (12 × 11)/(2 × 1) = 132/2 = 66</p><p><strong>Verification:</strong> This counts all ways to place 2 dividers among 10 stars, creating 3 groups for x, y, z respectively.</p><p>∴ Answer: <strong>66</strong></p>
Correct Answer: 66