Permutations & Combinations
Combinations
Grade 11
Question:
<p>If there are six straight lines in a plane, no two of which are parallel and no three of which pass through the same point, then find the number of points in which these lines intersect.</p>
Step-by-Step Solution
Key Concept: Each pair of non-parallel lines intersects at exactly one point. Since no three lines are concurrent, every pair of lines creates a unique intersection point. Simply count all possible pairs of 6 lines using combinations.
<p><strong>Step 1:</strong> Identify what creates intersection points. Two non-parallel lines intersect at exactly one point.</p><p><strong>Step 2:</strong> Since no two lines are parallel, every pair of lines will intersect.</p><p><strong>Step 3:</strong> Since no three lines pass through the same point, each pair of lines creates a unique intersection point (no point is shared by more than 2 lines).</p><p><strong>Step 4:</strong> Count the number of ways to choose 2 lines from 6 lines:</p><p>Number of intersection points = C(6,2) = 6!/(2!(6-2)!) = (6 × 5)/2 = 15</p><p><strong>∴ Answer: 15</strong></p>
Correct Answer: 15