<p>There are 10 points on a plane of which 5 points are collinear. Also, no three of the remaining 5 points are collinear. Then find:</p><p><strong>(i)</strong> the number of straight lines joining these points.</p><p><strong>(ii)</strong> the number of triangles formed by joining these points.</p>
Step-by-Step Solution
Key Concept: When points are collinear, they form ONE line instead of multiple lines. Use total combinations minus the 'overcounting' from collinear points, then add back the single line they actually form.
<p><strong>Step 1 (Straight Lines):</strong> Total lines from 10 points if no three were collinear = ¹⁰C₂ = 45</p><p><strong>Step 2:</strong> The 5 collinear points would normally give ⁵C₂ = 10 lines, but they actually form only 1 line.</p><p><strong>Step 3:</strong> Lines = ¹⁰C₂ - ⁵C₂ + 1 = 45 - 10 + 1 = <strong>36</strong></p><p><strong>Step 4 (Triangles):</strong> Total triangles from 10 points if no three collinear = ¹⁰C₃</p><p><strong>Step 5:</strong> Any 3 points chosen from the 5 collinear points do NOT form a triangle (they're collinear). Number of such invalid sets = ⁵C₃</p><p><strong>Step 6:</strong> Valid triangles = ¹⁰C₃ - ⁵C₃ = 120 - 10 = <strong>110</strong></p><p>∴ Answer: (i) 36 lines, (ii) ¹⁰C₃ - ⁵C₃ (or 110 triangles)</p>
Correct Answer: (i) 36, (ii) ¹⁰C₃ - ⁵C₃