Permutations & Combinations
Intersection points formula
Grade 11
Question:
<p>If there are <i>n</i> points in a plane such that no three are collinear, and <i>p</i> = <i>n</i>C₂, then find the total number of points of intersection of the lines joining these <i>n</i> points.</p>
Step-by-Step Solution
Key Concept: Use the combination formula twice: first to count lines from points, then to count intersections of those lines.
<p><strong>Step 1:</strong> We have <i>n</i> points in a plane with no three collinear.</p><p><strong>Step 2:</strong> The number of lines joining these <i>n</i> points is:</p><p>$$p = \binom{n}{2}$$</p><p><strong>Step 3:</strong> Since no three points are collinear, every pair of these lines will intersect at a unique point (except those sharing a common point from the original <i>n</i> points).</p><p><strong>Step 4:</strong> The total number of intersection points of these lines is:</p><p>$$\binom{p}{2} = \binom{\binom{n}{2}}{2}$$</p>
Correct Answer: pC2