Final Answer:
1296 rectangles.
On an 8x8 chessboard, the total number of rectangles with sides parallel to the borders is 1296, calculated by choosing 2 horizontal lines out of 9 and 2 vertical lines out of 9 using combinatorics.
Step-by-step explanation:
On an 8x8 chessboard, the total number of rectangles with sides parallel to the borders can be calculated using combinatorics. There are 9 horizontal lines and 9 vertical lines on the chessboard. To form a rectangle, you need to choose two horizontal lines and two vertical lines.
The number of ways to choose 2 horizontal lines out of 9 is denoted as 9C2, and similarly, the number of ways to choose 2 vertical lines out of 9 is 9C2. The total number of rectangles is the product of these combinations: 9C2 * 9C2 = 36 * 36 = 1296.
Understanding the principles of combinations is crucial in solving such problems. Combinations represent the different ways elements can be selected without regard to the order in which they are arranged. In this case, it's about selecting lines to form rectangles, and the formula for combinations captures this concept accurately.