<p>Find number of positive integral solutions of the equation \(x + y + z = 12\).</p>
Step-by-Step Solution
Key Concept: Use stars and bars theorem: distributing n identical items into k distinct groups equals C(n+k-1, k-1). Here we need C(12+3-1, 3-1) = C(14, 2).
<p><strong>Step 1:</strong> Recognize this is a problem of distributing 12 identical units among 3 distinct variables (x, y, z) where each must be a positive integer (≥ 1).</p><p><strong>Step 2:</strong> Since we need positive integers, each variable must get at least 1. Transform: let x = x' + 1, y = y' + 1, z = z' + 1 where x', y', z' ≥ 0. Then (x'+1) + (y'+1) + (z'+1) = 12, giving x' + y' + z' = 9.</p><p><strong>Step 3:</strong> Now distribute 9 identical units among 3 variables with no restrictions. By stars and bars formula: C(n+k-1, k-1) = C(9+3-1, 3-1) = C(11, 2).</p><p><strong>Step 4:</strong> Calculate C(11, 2) = (11 × 10)/(2 × 1) = 110/2 = 55.</p><p>∴ Answer: <strong>55</strong></p>
Correct Answer: 55