Carcass wrote:
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
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-coordinatesSo, 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)
Answer: C
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Brent Hanneson - founder of Greenlight Test Prep