[email protected] 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?
(Note that two rectangles that have the same four vertices that are labeled differently are considered to be the same rectangle
Notice that, if the rectangle is parallel to the x- and y-axes, then the coordinates of the 4 vertices will be such that:
- 2 vertices share one of the x-coordinates
- 2 vertices share the other x-coordinate
- 2 vertices share one of the y-coordinates
- 2 vertices share the other y-coordinate
For example, the points (8, -2), (11, -2), (8, 4) and (11, 4) create a rectangle AND they meet the above criteria.
So, to create a rectangle, all we need to do is select two x-coordinates and two y-coordinates.
Okay, now my solution . . .
Take the task of building rectangles and break it into stages.
Stage 1: Choose the two x-coordinates
The x-coordinates must be selected from {3,4,5,6,7,8,9,10,11}
Since the order of the selections does not matter, we can use combinations.
We can select 2 coordinates from 9 coordinates in 9C2 ways (
36 ways).
Aside: If anyone is interested, we have a free video on calculating combinations (like 9C2) in your head: https://www.gmatprepnow.com/module/gmat-counting?id=789
Stage 2: Choose the two y-coordinates
The y-coordinates must be selected from {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}
Since the order of the selections does not matter, we can use combinations.
We can select 2 coordinates from 11 coordinates in 11C2 ways (
55 ways).
By the Fundamental Counting Principle (FCP) we can complete the 2 stages (and build a rectangle) in
(36)(55) ways ([spoiler]= 1980 ways[/spoiler])
Cheers,
Brent
Aside: For more information about the FCP, we have a free video on the subject:
https://www.gmatprepnow.com/module/gmat-counting?id=775
