There are 27 points in plane. 5, 10 and 15 points are collinear on distinct lines. By joining these points, how many distinct lines can be formed?
Hint 1: Find the total number of lines formed by joining any 2 of the 27 points without constraints: .
Hint 2: Subtract the overcounted lines formed by the collinear groups: , , and .
Hint 3: Add back exactly 3 lines, since each of the three collinear groups still forms 1 valid distinct line.
Step 1 (Total Pairs of Points): There are points in a plane. Choosing any points forms a unique line. The total number of possible pairs of points is:
Step 2 (Subtract Overcounts due to Collinear Lines): Since groups of points are collinear, choosing any 2 points from the same collinear group does not form a new line — they all lie on the same single line.
Group 1: 5 collinear points on line 1. These 5 points form pairs which overcount the single line. We subtract these 10 pairs and add 1 line back.
Group 2: 10 collinear points on line 2. These 10 points form pairs. We subtract these 45 pairs and add 1 line back.
Group 3: 15 collinear points on line 3. These 15 points form pairs. We subtract these 105 pairs and add 1 line back.
Step 3 (Calculate Total Distinct Lines):
Step 4 (Conclusion): By joining these points, exactly 194 distinct lines can be formed.
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