Rectangle ABCD is constructed in the coordinate plane parallel to the x and yaxes. If the x and ycoordinates 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
Answer: C
Source: Grockit
Rectangle ABCD is constructed in the coordinate plane parallel
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First notice that, to construct this rectangle, the vertices will share several points.BTGModeratorVI wrote: ↑Sun Jul 19, 2020 1:38 pmRectangle ABCD is constructed in the coordinate plane parallel to the x and yaxes. If the x and ycoordinates 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
Answer: C
Source: Grockit
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 xcoordinates (2 and 9) and only 2 different ycoordinates (3 and 5)
So, to create the desired rectangle, we need only choose 2 different xcoordinates and 2 different ycoordinates
So, let's take the task of creating rectangles and break it into STAGES
STAGE 1: Select the 2 xcoordinates
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 ycoordinates
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)
Answer: C
As the rectangle is parallel to coordinate axes, the coordinates of the points of the rectangle would beBTGModeratorVI wrote: ↑Sun Jul 19, 2020 1:38 pmRectangle ABCD is constructed in the coordinate plane parallel to the x and yaxes. If the x and ycoordinates 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
Answer: C
Source: Grockit
\((X_1, Y_1), (X_2, Y_1), (X_2, Y_2), (X_1,Y_2)\)
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