<p>Find the number of non-negative integral solutions of \(x + y + z + w \leq 20\).</p>
Step-by-Step Solution
Key Concept: Transform the inequality x + y + z + w ≤ 20 into an equality by introducing a slack variable s ≥ 0, giving x + y + z + w + s = 20. Now count non-negative integer solutions using stars and bars.
<p><strong>Step 1:</strong> Convert the inequality to an equality by introducing slack variable s ≥ 0:</p><p>x + y + z + w ≤ 20 becomes x + y + z + w + s = 20, where s ≥ 0</p><p><strong>Step 2:</strong> Now we need non-negative integer solutions to x + y + z + w + s = 20 with 5 variables.</p><p><strong>Step 3:</strong> Apply Stars and Bars formula: The number of ways to distribute n identical objects into k distinct boxes is C(n+k-1, k-1).</p><p>Here, n = 20 (the sum) and k = 5 (variables: x, y, z, w, s).</p><p><strong>Step 4:</strong> Number of solutions = C(20+5-1, 5-1) = C(24, 4) = ²⁴C₄</p>
Correct Answer: ²⁴C₄