## Rectangle ABCD is constructed in the coordinate plane parallel

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### Rectangle ABCD is constructed in the coordinate plane parallel

by BTGModeratorVI » Sun Jul 19, 2020 1:38 pm

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Rectangle ABCD is constructed in the coordinate plane parallel to the x- and y-axes. If the x- and y-coordinates of each of the points are integers which satisfy 3 ≤ x ≤ 11 and -5 ≤ y ≤ 5, how many possible ways are there to construct rectangle ABCD?

396
1260
1980
7920
15840

Source: Grockit

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### Re: Rectangle ABCD is constructed in the coordinate plane parallel

by [email protected] » Thu Oct 21, 2021 9:35 am
BTGModeratorVI wrote:
Sun Jul 19, 2020 1:38 pm
Rectangle ABCD is constructed in the coordinate plane parallel to the x- and y-axes. If the x- and y-coordinates of each of the points are integers which satisfy 3 ≤ x ≤ 11 and -5 ≤ y ≤ 5, how many possible ways are there to construct rectangle ABCD?

396
1260
1980
7920
15840

Source: Grockit
First notice that, to construct this rectangle, the vertices will share several points.
For example, if the 4 vertices are at (2, 5), (2, -3), (9, 5) and (9, -3), then we get a rectangle.

Notice that there are only 2 different x-coordinates (2 and 9) and only 2 different y-coordinates (-3 and 5)

So, to create the desired rectangle, we need only choose 2 different x-coordinates and 2 different y-coordinates

So, let's take the task of creating rectangles and break it into STAGES

STAGE 1: Select the 2 x-coordinates
We can choose 2 values from the set {3, 4, 5, 6, 7, 8, 9, 10, and 11}
In other words, we must choose 2 of the 9 values in the set
Since the order in which we choose the numbers does not matter, we can use COMBINATIONS
We can select 2 number from 9 numbers in 9C2 ways (= 36 ways)

STAGE 2: Select the 2 y-coordinates
We can choose 2 values from the set {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}
In other words, we must choose 2 of the 11 values in the set
We can select 2 number from 11 numbers in 11C2 ways (= 55 ways)

By the Fundamental Counting Principle (FCP), we can complete the 2 stages (and thus create a rectangle) in (36)(55) ways (= 1980 ways)

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### Re: Rectangle ABCD is constructed in the coordinate plane parallel

by swerve » Thu Oct 21, 2021 2:25 pm
BTGModeratorVI wrote:
Sun Jul 19, 2020 1:38 pm
Rectangle ABCD is constructed in the coordinate plane parallel to the x- and y-axes. If the x- and y-coordinates of each of the points are integers which satisfy 3 ≤ x ≤ 11 and -5 ≤ y ≤ 5, how many possible ways are there to construct rectangle ABCD?

396
1260
1980
7920
15840

Source: Grockit
As the rectangle is parallel to coordinate axes, the coordinates of the points of the rectangle would be

$$(X_1, Y_1), (X_2, Y_1), (X_2, Y_2), (X_1,Y_2)$$

Given that $$X_1, X_2$$ lie between $$3$$ and $$11$$ ie., $$9$$ possible numbers

Possible combinations for $$X_1, X_2$$ would be $$9C2 = 36$$

Similarly, Possible combinations for $$Y_1, Y_2$$ would be $$11C2 = 55$$

Possible ways of constructing rectangle is by selecting any of the combination of $$X_1, X_2$$ and $$Y_1, Y_2$$

$$= 36 \ast 55 = 1980$$

Therefore, C

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