<p>There are 16 points in a plane of which 8 are on one line and 8 are on another line. The number of triangles that can be formed with the points as vertices is ______.</p>
Step-by-Step Solution
Key Concept: Triangles require 3 non-collinear points. You must subtract cases where all 3 points are collinear (from the 8 points on each line) from the total ways to choose 3 points.
<p><strong>Step 1:</strong> Total ways to choose 3 points from 16 points = C(16,3) = 16×15×14/(3×2×1) = 560</p><p><strong>Step 2:</strong> Subtract collinear triples from Line 1: C(8,3) = 8×7×6/(3×2×1) = 56</p><p><strong>Step 3:</strong> Subtract collinear triples from Line 2: C(8,3) = 56</p><p><strong>Step 4:</strong> Valid triangles = C(16,3) - C(8,3) - C(8,3) = 560 - 56 - 56 = 448</p><p>∴ Answer: <strong>448</strong></p>
Correct Answer: 448