Permutations & Combinations
Counting Rectangles in Grid
Grade 11
Question:
<p>There is a rectangular sheet of dimension \((2m - 1) \times (2n - 1)\), (where \(m > 0, n > 0\)). It has been divided into squares of unit area by drawing lines perpendicular to the sides. Find the number of rectangles having sides of odd unit length.</p>
<p>(a) \((m + n + 1)^2\)</p>
<p>(b) \(mn(m + 1)(n + 1)\)</p>
<p>(c) \(4m + n - 2\)</p>
<p>(d) \(m^2n^2\)</p>
Step-by-Step Solution
Key Concept: A rectangle with odd side lengths is formed by selecting two vertical lines and two horizontal lines from positions corresponding to odd-length intervals. Count the number of ways to choose such pairs.
<p>A rectangle with odd side lengths requires selecting odd number of unit segments in both horizontal and vertical directions. In a $(2m-1) \times (2n-1)$ grid, there are $m$ odd-length horizontal segments and $n$ odd-length vertical segments. To form a rectangle of odd side length requires choosing 2 points from consecutive odd segments, giving $m^2$ choices horizontally and $n^2$ choices vertically.</p><p>∴ Answer is $m^2n^2$.</p>
Correct Answer: D