<p>Find the number of positive integral solutions satisfying the equation \((x_1 + x_2 + x_3)(y_1 + y_2) = 77\).</p>
Step-by-Step Solution
Key Concept: Since 77 = 7 × 11, we need to find all factor pairs (a,b) where a·b = 77, then count positive integer solutions: partitioning a into 3 parts and b into 2 parts using stars and bars.
<p><strong>Step 1: Find factor pairs of 77</strong></p><p>77 = 1 × 77 = 7 × 11</p><p>Possible pairs (a,b) where a = x₁ + x₂ + x₃ and b = y₁ + y₂:</p><p>(1, 77), (7, 11), (11, 7), (77, 1)</p><p><strong>Step 2: Apply stars and bars for positive integers</strong></p><p>For x₁ + x₂ + x₃ = a (positive integers): C(a-1, 2) ways</p><p>For y₁ + y₂ = b (positive integers): C(b-1, 1) ways</p><p><strong>Step 3: Calculate for each case</strong></p><p>• (a,b) = (1,77): C(0,2) × C(76,1) = 0 (impossible: can't write 1 as sum of 3 positive integers)</p><p>• (a,b) = (7,11): C(6,2) × C(10,1) = 15 × 10 = 150</p><p>• (a,b) = (11,7): C(10,2) × C(6,1) = 45 × 6 = 270</p><p>• (a,b) = (77,1): C(76,2) × C(0,1) = ? × 0 = 0 (impossible: can't write 1 as sum of 2 positive integers)</p><p><strong>Step 4: Sum all cases</strong></p><p>Total = 150 + 270 = 420</p><p>∴ Answer: <strong>420</strong></p>
Correct Answer: 420