<p>There are red, green and white identical balls, each being 10 in number. The number of selections of 10 balls in which the number of red balls is double the number of green balls is ______.</p>
Step-by-Step Solution
Key Concept: Set up the constraint that red balls = 2 × green balls, then use non-negative integer solutions to find valid distributions of the third color (white balls). The key is recognizing this reduces to counting valid triplets (r, g, w) where r = 2g and r + g + w = 10.
<p><strong>Step 1:</strong> Let r = number of red balls, g = number of green balls, w = number of white balls.</p><p><strong>Step 2:</strong> We have the constraints:</p><ul><li>r = 2g (given condition)</li><li>r + g + w = 10 (total selection)</li><li>0 ≤ r ≤ 10, 0 ≤ g ≤ 10, 0 ≤ w ≤ 10</li></ul><p><strong>Step 3:</strong> Substitute r = 2g into the total: 2g + g + w = 10, so 3g + w = 10, giving w = 10 - 3g.</p><p><strong>Step 4:</strong> Apply constraints on g:</p><ul><li>From w ≥ 0: 10 - 3g ≥ 0 ⟹ g ≤ 10/3 ⟹ g ≤ 3</li><li>From r ≤ 10: 2g ≤ 10 ⟹ g ≤ 5</li><li>From g ≥ 0: g ≥ 0</li></ul><p><strong>Step 5:</strong> The binding constraint is g ≤ 3. Combined with g ≥ 0, we have g ∈ {0, 1, 2, 3}.</p><p><strong>Step 6:</strong> Verify each case:</p><ul><li>g = 0: r = 0, w = 10 ✓ (0, 0, 10)</li><li>g = 1: r = 2, w = 7 ✓ (2, 1, 7)</li><li>g = 2: r = 4, w = 4 ✓ (4, 2, 4)</li><li>g = 3: r = 6, w = 1 ✓ (6, 3, 1)</li></ul><p><strong>Step 7:</strong> Count the number of valid selections = 4</p><p>∴ Answer: <strong>4</strong></p>
Correct Answer: 4