In a comment, Steven McDonald already gave a nice solution to the puzzle here. My own solution is equivalent but framed a bit differently; I went for a more visual explanation.
Each of the rectangles we want to count is determined by some two rows and two columns of the 10x10 grid, as shown in this example:
For the rows we have 10 choices for the first, which leaves 9 choices for the second, but that will have double-counted each combination, because for example row 5 and row 9 represents the same pair as row 9 and row 5. So we must divide by 2 and there are
(10 x 9) / 2 = 45
distinct combinations.
Similar reasoning gives 45 of ways to choose two distinct columns and so we find there are 45 x 45 = 2025 different rectangles.
Solution to "A dotty Puzzle..."
In a comment, Steven McDonald already gave a nice solution to the puzzle here. My own solution is equivalent but framed a bit differently; I went for a more visual explanation.
Each of the rectangles we want to count is determined by some two rows and two columns of the 10x10 grid, as shown in this example:
For the rows we have 10 choices for the first, which leaves 9 choices for the second, but that will have double-counted each combination, because for example row 5 and row 9 represents the same pair as row 9 and row 5. So we must divide by 2 and there are
(10 x 9) / 2 = 45
distinct combinations.
Similar reasoning gives 45 of ways to choose two distinct columns and so we find there are 45 x 45 = 2025 different rectangles.