Permutations & Combinations
Geometric counting
Grade 11
Question:
<p>If \(m\) parallel lines in a plane are intersected by a family of \(n\) parallel lines, the number of parallelograms that can be formed is</p>
<p>\(\dfrac{1}{4}mn(m-1)(n-1)\)</p>
<p>\(\dfrac{1}{2}mn(m-1)(n-1)\)</p>
<p>\(\dfrac{1}{4}m^2n^2\)</p>
<p>none of these</p>
Step-by-Step Solution
Key Concept: A parallelogram requires exactly 2 lines from the first family and 2 lines from the second family. The number of ways to choose these pairs independently gives the total count.
<p><strong>Step 1:</strong> Understand the geometric setup. We have m parallel lines in one direction and n parallel lines in another direction (not parallel to the first family).</p><p><strong>Step 2:</strong> A parallelogram is formed by selecting exactly 2 lines from the m parallel lines AND exactly 2 lines from the n parallel lines.</p><p><strong>Step 3:</strong> Number of ways to choose 2 lines from m parallel lines = C(m,2) = m(m-1)/2</p><p><strong>Step 4:</strong> Number of ways to choose 2 lines from n parallel lines = C(n,2) = n(n-1)/2</p><p><strong>Step 5:</strong> By the fundamental counting principle, total number of parallelograms = C(m,2) × C(n,2) = [m(m-1)/2] × [n(n-1)/2]</p><p><strong>Step 6:</strong> Simplify: = m(m-1)n(n-1)/4</p><p>∴ Answer: <strong>C(m,2) × C(n,2) = m(m-1)n(n-1)/4</strong></p>
Correct Answer: A