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IIn 6 x 9 grid, how many rectangles that consists of an even number of unit squares?

 Jun 17, 2023
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There are 120 rectangles that consists of an even number of unit squares in a 6 x 9 grid.

To solve this, we can use the following steps:

Find the total number of rectangles that can be formed in a 6 x 9 grid. This can be done by multiplying the number of horizontal lines (6) by the number of vertical lines (9), which gives us 54 rectangles.

Subtract the number of rectangles that contain an odd number of unit squares. This can be done by finding the number of rectangles that can be formed by choosing two horizontal lines and two vertical lines such that the difference between the horizontal lines is odd and the difference between the vertical lines is odd. There are 18 ways to choose two horizontal lines such that the difference is odd, and there are 36 ways to choose two vertical lines such that the difference is odd. This gives us a total of 64 rectangles.

Subtract the number of rectangles that contain both an odd number of horizontal units and an odd number of vertical units. There is only one way to choose two horizontal lines and two vertical lines such that both the difference between the horizontal lines and the difference between the vertical lines is odd. This gives us one rectangle.

The total number of rectangles that consists of an even number of unit squares is 54 - 64 + 1 = 120.

I hope this helps!

 Jun 17, 2023

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