How to Count Rectangles in a Grid: A Combinatorial Approach

The Rectangle Counting Problem

• 💡 The problem asks to count the number of rectangles in a diagram with two rows and three columns of rectangles, plus an irrelevant triangle.
• 🎯 The triangle attached to the end of the grid is irrelevant to the rectangle count and can be ignored.

Manual Counting Method

• 🔢 A manual count involves identifying and summing rectangles of different sizes:
  • 1x1: 6 rectangles
  • 1x2: 4 rectangles
  • 2x1: 3 rectangles
  • 2x2: 2 rectangles
  • 1x3: 2 rectangles
  • 2x3: 1 rectangle
• ✅ The sum of these counts is 6 + 4 + 3 + 2 + 2 + 1 = 18 rectangles.
• ⚠️ While this method yields an answer, it's prone to errors and lacks certainty.

Combinatorial Approach for General Grids

• 🧠 A more robust method uses combinatorics: a rectangle is defined by choosing two distinct horizontal lines and two distinct vertical lines.
• 📈 For a grid with h horizontal lines and v vertical lines, the number of rectangles is (h choose 2) * (v choose 2).
• 🚀 This formula is derived from the number of ways to select pairs of lines that form the boundaries of a rectangle.

Applying the Formula to the Original Problem

• 📏 The given grid has 3 horizontal lines and 4 vertical lines.
• 🧮 Applying the formula: (3 choose 2) * (4 choose 2) = 3 * 6 = 18 rectangles.
• 💯 This combinatorial method provides a mathematically certain answer, confirming the manual count and offering a scalable solution for larger grids.