Question

STATEMENT - 1 : Number of rectangles on chessboard is \({ }^{8} \mathrm{C}_{2} \cdot{ }^{8} \mathrm{C}_{2}\) STATEMENT - 2 : To form a rectangle we have to select any two of the horizontal lines and any two of the vertical lines. (1) Statement \(-1\) is True, Statement \(-2\) is True ; Statement \(-2\) is a correct explanation for Statement \(-1\) (2) Statement-1 is True, Statement-2 is True ; Statement-2 is NOT a correct explanation for Statement-1 (3) Statement \(-1\) is True, Statement \(-2\) is False (4) Statement \(-1\) is False, Statement \(-2\) is True

Short answer

(1) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1.

Step-by-step solution

Understand the Problem

Analyze the statements given. Statement-1 discusses the number of rectangles on a chessboard using combinations. Statement-2 provides an explanation on how to form a rectangle by selecting two horizontal and two vertical lines.

Analyze Statement-1

Statement-1 claims that the number of rectangles on a chessboard can be calculated using the formula \({ }^{8} \text{C}_{2} \cdot{ }^{8} \text{C}_{2}\). This is because to form a rectangle, you need to select 2 out of the 8 horizontal lines and 2 out of the 8 vertical lines. Hence, Statement-1 is correct.

Analyze Statement-2

Statement-2 explains that to form a rectangle, one needs to select any two horizontal lines and any two vertical lines. This is indeed the correct logical explanation for how the formula in Statement-1 is derived. Thus, Statement-2 is also correct.

Verify Explanation

Since Statement-2 correctly explains the logic used in Statement-1, it is a correct explanation. Hence, both Statement-1 is True, Statement-2 is True and Statement-2 correctly explains Statement-1.

Key concepts

Combinatorial Mathematics

Combinatorial mathematics is the branch of math that deals with counting, arranging, and selecting objects. In the context of the chessboard problem, we use combinatorics to determine the number of rectangles. Specifically, we use the combination formula to count how we can pick sets of lines. The formula \text{C(n, k)} = \frac{n!}{k!(n-k)!}\ is used to calculate combinations, which tell us how many ways we can pick \( k \) items from a set of \( n \) items without considering the order.

For the chessboard with 8 rows and 8 columns, the problem asks us to select 2 out of 8 horizontal lines and 2 out of 8 vertical lines. By using the combination formula for both selections, we get \({ }^{8} \text{C}_{2} \). Therefore, the total number of rectangles is calculated by multiplying the two combinations: \({ }^{8} \text{C}_{2} \cdot{ }^{8} \text{C}_{2} \). This ensures we have considered all possible selections of horizontal and vertical lines to form rectangles. The answer is \text{C(8, 2)} \cdot \text{C(8, 2)} = 28 \cdot 28 = 784\ rectangles.

Geometry on Chessboard

To understand the geometry involved, we need to visualize the chessboard not just as a grid of squares but as a grid of intersecting lines. There are 9 horizontal and 9 vertical lines on a chessboard. A rectangle is formed by selecting two different horizontal lines and two different vertical lines. This selection demarcates the four sides of a rectangle.

Visually, imagine selecting any two horizontal lines from the top, and any two vertical lines from the left. These points where lines intersect become the corners of the rectangle. By counting all unique ways to pick these lines, we organically arrive at the result derived through combinatorial mathematics.

This geometric approach helps solidify the understanding that we are not just selecting squares, but instead working with the lines that make up the chessboard grid.

Probability and Statistics

Probability and statistics often intersect with combinatorial problems like the number of rectangles in a chessboard. Even though the specific task doesn't require calculating probability, it's a useful concept to explore.

In this scenario, the probability approach could ask: what is the likelihood of forming a certain type of rectangle? First, we understand there are 784 possible rectangles. If we wanted to find the probability of selecting any one rectangle, it would be one in 784 or \( \frac{1}{784} \).

Moreover, statistics helps us understand the distribution of different shapes and sizes of rectangles on the chessboard. This can deepen our understanding and provide new insights. For instance, rectangles can vary in dimensions, which adds layers to analyzing outcomes but ultimately strengthens combinatorial methods by applying real-world probability and statistical analysis.