<p>Find the number of non-negative integral solutions of \(x_1 + x_2 + x_3 + 4x_4 = 20\).</p>
Step-by-Step Solution
Key Concept: Decompose the constraint by treating 4x₄ as a single unit: first determine valid values of x₄ (0 to 5), then for each x₄, count solutions to x₁ + x₂ + x₃ = 20 - 4x₄ using stars and bars.
<p><strong>Step 1:</strong> Recognize that x₄ can only take values where 4x₄ ≤ 20, so x₄ ∈ {0, 1, 2, 3, 4, 5}.</p><p><strong>Step 2:</strong> For each value of x₄, we need non-negative solutions to x₁ + x₂ + x₃ = 20 - 4x₄.</p><p><strong>Step 3:</strong> Using stars and bars, the number of solutions is C(20 - 4x₄ + 2, 2) = C(22 - 4x₄, 2).</p><p><strong>Step 4:</strong> Sum over all valid values of x₄:</p><p>• x₄ = 0: C(22, 2) = 231</p><p>• x₄ = 1: C(18, 2) = 153</p><p>• x₄ = 2: C(14, 2) = 91</p><p>• x₄ = 3: C(10, 2) = 45</p><p>• x₄ = 4: C(6, 2) = 15</p><p>• x₄ = 5: C(2, 2) = 1</p><p><strong>Step 5:</strong> Total = 231 + 153 + 91 + 45 + 15 + 1 = <strong>536</strong></p>
Correct Answer: 536