<p>Find the maximum number of points of intersection of 7 straight lines and 5 circles when 3 straight lines are parallel and 2 circles are concentric.</p>
Step-by-Step Solution
Key Concept: Maximum intersections occur when objects are in general position (no unwanted coincidences). Count all possible pairs and subtract intersections that are forbidden due to the given constraints (parallel lines don't meet, concentric circles don't intersect).
<p><strong>Step 1: Calculate maximum intersections without restrictions</strong></p><p>• Line-Line: C(7,2) × 1 = 21 points</p><p>• Circle-Circle: C(5,2) × 2 = 20 points</p><p>• Line-Circle: 7 × 5 × 2 = 70 points</p><p>Total without restrictions = 21 + 20 + 70 = 111 points</p><p><strong>Step 2: Subtract intersections lost due to 3 parallel lines</strong></p><p>• 3 parallel lines among themselves intersect at 0 points instead of C(3,2) = 3 points</p><p>Loss = 3 points</p><p><strong>Step 3: Subtract intersections lost due to 2 concentric circles</strong></p><p>• 2 concentric circles intersect at 0 points instead of 2 points</p><p>Loss = 2 points</p><p><strong>Step 4: Apply principle of inclusion-exclusion</strong></p><p>Maximum intersection points = 111 - 3 - 2 = <strong>106</strong></p>
Correct Answer: 106