Permutations & Combinations
Geometric counting
Grade 11
Question:
<p>Straight lines are drawn by joining \(m\) points on a straight line to \(n\) points on another line. Then excluding the given points, the number of point of intersections of the lines drawn is (no two lines drawn are parallel and no three lines are concurrent)</p>
<p>\(\dfrac{1}{4}mn(m-1)(n-1)\)</p>
<p>\(\dfrac{1}{2}mn(m-1)(n-1)\)</p>
<p>\(\dfrac{1}{2}m^2n^2\)</p>
<p>\(\dfrac{1}{4}m^2n^2\)</p>
Step-by-Step Solution
Key Concept: Each pair of lines intersects at exactly one point inside the region between the two given lines. A line is determined by one point from the first set and one point from the second set, so the number of intersection points equals the number of ways to choose 2 lines from all possible lines, which equals C(mn,2) minus the collinear intersections that fall on the original lines.
<p><strong>Step 1:</strong> Count total lines drawn. We join m points on line 1 to n points on line 2, giving us m×n lines total.</p><p><strong>Step 2:</strong> Any two lines will intersect at exactly one point (given no two are parallel and no three concurrent). Total intersection points from C(mn,2) = mn(mn-1)/2.</p><p><strong>Step 3:</strong> Subtract intersections that occur ON the original lines. Lines through the same point on line 1 all pass through that point (these are m groups of n lines each). Similarly for line 2 (n groups of m lines each).</p><p><strong>Step 4:</strong> From the m points on line 1: At each point, n lines meet. Number of pairs = m·C(n,2) = m·n(n-1)/2</p><p><strong>Step 5:</strong> From the n points on line 2: At each point, m lines meet. Number of pairs = n·C(m,2) = n·m(m-1)/2</p><p><strong>Step 6:</strong> Interior intersection points = mn(mn-1)/2 - m·n(n-1)/2 - n·m(m-1)/2</p><p>= mn/2[mn - 1 - (n-1) - (m-1)]</p><p>= mn/2[mn - n - m + 1]</p><p>= <strong>mn(m-1)(n-1)/2</strong></p><p>∴ Answer: A</p>
Correct Answer: A