Number of straight lines from 10 points with 7 collinear: How many distinct straight lines can be formed?
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A26
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B21
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C25
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DNone of these
Answer
Correct Answer: 25
Explanation
Introduction / Context:Distinct lines arise from pairs of points, but when many points are collinear, all pairs among them correspond to the same line and must not be overcounted.
Given Data / Assumptions:10 points total; 7 on one line; assume the other 3 are in general position relative to the 7 and each other.
Concept / Approach:Total potential lines from all pairs minus the overcount among the collinear 7, then add back that single line once.
Step-by-Step Solution:
Total pairs = C(10, 2) = 45Pairs along the 7-point line = C(7, 2) = 21Distinct lines = 45 − 21 + 1 = 25Verification / Alternative check:List categories: the one big line from the 7, lines from an off-line point to each of the 7, and lines among off-line points — totals reconcile to 25.
Why Other Options Are Wrong:26 or 21 result from missing either the subtraction or the add-back of one line.
Common Pitfalls:Forgetting to add the unique collinear line back after subtracting its pairs.
Final Answer:25