<p>A box contains 5 different red and 6 different white balls. In how many ways can 6 balls be selected so that there are at least two balls of each color?</p>
Step-by-Step Solution
Key Concept: With the constraint of at least 2 balls of each color from a selection of 6 balls, the possible distributions are limited to (2R,4W), (3R,3W), and (4R,2W). Calculate each case using combinations and sum them.
<p><strong>Step 1:</strong> Identify valid distributions. We need 6 balls total with at least 2 red AND at least 2 white.</p><p>Valid cases: (2R,4W), (3R,3W), (4R,2W)</p><p><strong>Step 2:</strong> Calculate Case 1 (2 red, 4 white):</p><p>C(5,2) × C(6,4) = 10 × 15 = 150</p><p><strong>Step 3:</strong> Calculate Case 2 (3 red, 3 white):</p><p>C(5,3) × C(6,3) = 10 × 20 = 200</p><p><strong>Step 4:</strong> Calculate Case 3 (4 red, 2 white):</p><p>C(5,4) × C(6,2) = 5 × 15 = 75</p><p><strong>Step 5:</strong> Add all cases: 150 + 200 + 75 = 425</p><p>∴ <strong>Answer: 425</strong></p>
Correct Answer: 425