<p>Suppose that 20 pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number of beams is __________.</p>
Step-by-Step Solution
Key Concept: Each pillar connects to all non-adjacent pillars. With 20 pillars arranged in a circle, each pillar is adjacent to exactly 2 others (neighbors), so connects to 20 - 1 - 2 = 17 pillars. The total count 20 × 17 must be divided by 2 since each beam connects two pillars.
<p><strong>Step 1:</strong> Identify the constraint. With 20 pillars on a circle's boundary, each pillar has exactly 2 adjacent pillars (its immediate neighbors clockwise and counterclockwise).</p><p><strong>Step 2:</strong> Count connections per pillar. Each pillar connects to all non-adjacent pillars: 20 - 1 (itself) - 2 (adjacent) = 17 pillars.</p><p><strong>Step 3:</strong> Calculate total beams. If we count from each pillar: 20 × 17 = 340. Since each beam connects two pillars (counted twice in the above), divide by 2: 340 ÷ 2 = 170.</p><p><strong>Alternative approach:</strong> Total possible connections = C(20,2) = 190. Subtract adjacent pairs = 20. Therefore, non-adjacent connections = 190 - 20 = 170.</p><p>∴ Answer: <strong>170</strong></p>
Correct Answer: 170