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 â‰¤ x â‰¤ 11 and 5 â‰¤ y â‰¤ 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
This topic has expert replies

 Master  Next Rank: 500 Posts
 Posts: 429
 Joined: 19 Sep 2012
 Thanked: 6 times
 Followed by:4 members
GMAT/MBA Expert
 Brent@GMATPrepNow
 GMAT Instructor
 Posts: 13519
 Joined: 08 Dec 2008
 Location: Vancouver, BC
 Thanked: 5254 times
 Followed by:1256 members
 GMAT Score:770
Notice that, if the rectangle is parallel to the x and yaxes, then the coordinates of the 4 vertices will be such that:shibsriz@gmail.com wrote: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 â‰¤ x â‰¤ 11 and 5 â‰¤ y â‰¤ 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
 2 vertices share one of the xcoordinates
 2 vertices share the other xcoordinate
 2 vertices share one of the ycoordinates
 2 vertices share the other ycoordinate
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 xcoordinates and two ycoordinates.
Okay, now my solution . . .
Take the task of building rectangles and break it into stages.
Stage 1: Choose the two xcoordinates
The xcoordinates 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/gmatcounting?id=789
Stage 2: Choose the two ycoordinates
The ycoordinates 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/gmatcounting?id=775
Brent Hanneson  Creator of GMATPrepNow.com
Use my video course along with Beat The GMAT's free 60Day Study Guide
Sign up for free Question of the Day emails
And check out all of these free resources
Use my video course along with Beat The GMAT's free 60Day Study Guide
Sign up for free Question of the Day emails
And check out all of these free resources
 vishugogo
 Master  Next Rank: 500 Posts
 Posts: 131
 Joined: 30 Aug 2011
 Location: India
 Thanked: 28 times
 Followed by:6 members
Dear Brent,
I was solving in another way but got stuck
I decided to solve for each coordinates
For A selection could be done in 9*11 ways
For B selection could be done in 8*11 ways
Now what??
I was solving in another way but got stuck
I decided to solve for each coordinates
For A selection could be done in 9*11 ways
For B selection could be done in 8*11 ways
Now what??
GMAT/MBA Expert
 Brent@GMATPrepNow
 GMAT Instructor
 Posts: 13519
 Joined: 08 Dec 2008
 Location: Vancouver, BC
 Thanked: 5254 times
 Followed by:1256 members
 GMAT Score:770
I like your approach.vishugogo wrote:Dear Brent,
I was solving in another way but got stuck
I decided to solve for each coordinates
For A selection could be done in 9*11 ways
For B selection could be done in 8*11 ways
Now what??
If you let A and B be points on the rectangle's diagonal, then a unique rectangle is defined by these two diagonal points. For example, the points (8, 2) and (11, 4) define the entire rectangle with points (8, 2), (11, 2), (8, 4) and (11, 4)
HOWEVER, this approach gets troublesome when we have to later go back and see how many duplicate rectangles we have created.
In your solution, you select point A (one of the vertices).
As you suggest, this can be done in (9)(11) ways (99 ways)
If point B is to be the point diagonal to point A, the x and ycoordinates of point B must be different from the x and ycoordinates of point A.
So, we can select point B in (8)(10) ways (80 ways)
So, the total number of ways to select points A and B = (99)(80) = 7920
HOWEVER, we have inadvertently counted some rectangles more than once.
For example, if we interchange labels for points A and B, we still get the same rectangle, so we have counted the same rectangle twice.
Also if we switch labels for points A and B with the labels for points C and D we still get the same rectangle as well, so we have actually counted the same rectangle 4 times.
So, we must divide 7920 by 4 to get 1980
Cheers,
Brent
Brent Hanneson  Creator of GMATPrepNow.com
Use my video course along with Beat The GMAT's free 60Day Study Guide
Sign up for free Question of the Day emails
And check out all of these free resources
Use my video course along with Beat The GMAT's free 60Day Study Guide
Sign up for free Question of the Day emails
And check out all of these free resources

 Junior  Next Rank: 30 Posts
 Posts: 25
 Joined: 27 Apr 2016
Hi Brent ,
I had a small question.
Like in this question you have used Fundamental counting Principle,which says that if a work has two tasks and one task can be done in x ways and another task can be done in y ways ,then the total no of ways to accomplish the work is x*y ways.
Now my question is ,"by total no of ways " do we mean "total UNIQUE ways" because in the below question you have not taken any added steps to remove duplicate rectangles.
Thanks for the help.
Regards
Karishma Duggal
I had a small question.
Like in this question you have used Fundamental counting Principle,which says that if a work has two tasks and one task can be done in x ways and another task can be done in y ways ,then the total no of ways to accomplish the work is x*y ways.
Now my question is ,"by total no of ways " do we mean "total UNIQUE ways" because in the below question you have not taken any added steps to remove duplicate rectangles.
Thanks for the help.
Regards
Karishma Duggal

 Junior  Next Rank: 30 Posts
 Posts: 25
 Joined: 27 Apr 2016
Brent@GMATPrepNow wrote:Notice that, if the rectangle is parallel to the x and yaxes, then the coordinates of the 4 vertices will be such that:shibsriz@gmail.com wrote: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 â‰¤ x â‰¤ 11 and 5 â‰¤ y â‰¤ 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
 2 vertices share one of the xcoordinates
 2 vertices share the other xcoordinate
 2 vertices share one of the ycoordinates
 2 vertices share the other ycoordinate
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 xcoordinates and two ycoordinates.
Okay, now my solution . . .
Take the task of building rectangles and break it into stages.
Stage 1: Choose the two xcoordinates
The xcoordinates 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/gmatcounting?id=789
Stage 2: Choose the two ycoordinates
The ycoordinates 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/gmatcounting?id=775
Hi Brent ,
I had a small question.
Like in this question you have used Fundamental counting Principle,which says that if a work has two tasks and one task can be done in x ways and another task can be done in y ways ,then the total no of ways to accomplish the work is x*y ways.
Now my question is ,"by total no of ways " do we mean "total UNIQUE ways" because in the below question you have not taken any added steps to remove duplicate rectangles.
Thanks for the help.
Regards
Karishma Duggal
GMAT/MBA Expert
 Brent@GMATPrepNow
 GMAT Instructor
 Posts: 13519
 Joined: 08 Dec 2008
 Location: Vancouver, BC
 Thanked: 5254 times
 Followed by:1256 members
 GMAT Score:770
Hi Karishma,karishma315 wrote: Hi Brent ,
I had a small question.
Like in this question you have used Fundamental counting Principle,which says that if a work has two tasks and one task can be done in x ways and another task can be done in y ways ,then the total no of ways to accomplish the work is x*y ways.
Now my question is ,"by total no of ways " do we mean "total UNIQUE ways" because in the below question you have not taken any added steps to remove duplicate rectangles.
Thanks for the help.
Regards
Karishma Duggal
There is no duplication with my solution. We select 2 xcoordinates and 2 ycoordinates. That's all.
Once we've selected the 4 values, there's only 1 possible rectangle that can be created.
For example, if we select 8 and 11 for the xcoordinates and we select 2 and 4 for the ycoordinates, the ONLY rectangle that can get created has vertices at(8, 2), (11, 2), (8, 4) and (11, 4)
Cheers,
Brent
Brent Hanneson  Creator of GMATPrepNow.com
Use my video course along with Beat The GMAT's free 60Day Study Guide
Sign up for free Question of the Day emails
And check out all of these free resources
Use my video course along with Beat The GMAT's free 60Day Study Guide
Sign up for free Question of the Day emails
And check out all of these free resources