<p>In how many ways can 6 boys and 4 girls sit in a row so that no boy is between two girls?</p>
Step-by-Step Solution
Key Concept: No boy between two girls means girls cannot be separated by boys—girls must form contiguous blocks, and boys fill remaining spaces. Treat each contiguous group of girls as a unit first, then arrange all units.
<p><strong>Step 1: Understand the constraint</strong><br/>No boy between two girls means if two girls are adjacent or separated, no boy can be between them. This forces all girls to sit together in one contiguous block (or isolated individually with no boys nearby, which is restrictive).</p><p><strong>Step 2: Arrange as units</strong><br/>Treat 4 girls as ONE single unit and 6 boys as individual units. This gives us 7 units total (1 girl-block + 6 boys).</p><p><strong>Step 3: Arrange the 7 units</strong><br/>The 7 units can be arranged in 7! ways.</p><p><strong>Step 4: Arrange girls within their block</strong><br/>The 4 girls within their unit can arrange themselves in 4! ways.</p><p><strong>Step 5: Arrange boys</strong><br/>The 6 boys can arrange themselves in 6! ways among themselves (this is already counted in unit arrangements).</p><p><strong>Step 6: Calculate total</strong><br/>Total arrangements = 7! × 4! = 5040 × 24 = <strong>120,960</strong></p><p>∴ Answer: <strong>7! × 4! = 120,960</strong></p>
Correct Answer: 7