<p>In how many ways can 6 boys and 6 girls be arranged in a row so that no two boys and no two girls sit together?</p>
Step-by-Step Solution
Key Concept: For alternating arrangements, fix one gender's positions first, then arrange the other gender in the remaining slots. The arrangement must strictly alternate: B-G-B-G... or G-B-G-B...
<p><strong>Step 1:</strong> For alternating arrangement with 6 boys and 6 girls in a row, we must have the pattern: B-G-B-G-B-G-B-G-B-G-B-G or G-B-G-B-G-B-G-B-G-B-G-B</p><p><strong>Step 2:</strong> For pattern starting with B: Boys occupy positions 1,3,5,7,9,11 (can be arranged in 6! ways) and girls occupy positions 2,4,6,8,10,12 (can be arranged in 6! ways). Total = 6! × 6!</p><p><strong>Step 3:</strong> For pattern starting with G: Girls occupy odd positions and boys occupy even positions. Total = 6! × 6!</p><p><strong>Step 4:</strong> Total arrangements = 2 × (6! × 6!) = 2 × 6! × 6! = 2 × 720 × 720 = 1,036,800</p><p><strong>Step 5:</strong> Simplified: Total = 2 × (6!)²</p><p>∴ Answer: <strong>2 × (6!)² or 1,036,800</strong></p>
Correct Answer: 2