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100 points for $49 worth of Veritas practice GMATs FREE VERITAS PRACTICE GMAT EXAMS Earn 10 Points Per Post Earn 10 Points Per Thanks Earn 10 Points Per Upvote ## wonderful P & C ques : tagged by: Brent@GMATPrepNow ##### This topic has 8 expert replies and 7 member replies Goto page • 1, • 2 ## wonderful P & C ques : 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 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.) 1. 396 2. 1260 3. 1980 4. 7920 5. 15840 ### GMAT/MBA Expert GMAT Instructor Joined 08 Dec 2008 Posted: 13046 messages Followed by: 1253 members Upvotes: 5254 GMAT Score: 770 ankitbagla 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 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.) 1. 396 2. 1260 3. 1980 4. 7920 5. 15840 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: http://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 (= 1980 ways = C) Cheers, Brent Aside: For more information about the FCP, we have a free video on the subject: http://www.gmatprepnow.com/module/gmat-counting?id=775 _________________ Brent Hanneson â€“ Creator of GMATPrepNow.com Use my video course along with Sign up for free Question of the Day emails And check out all of these free resources GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMATâ€™s FREE 60-Day Study Guide and reach your target score in 2 months! Master | Next Rank: 500 Posts Joined 19 Sep 2013 Posted: 269 messages Followed by: 7 members Upvotes: 94 Brent@GMATPrepNow wrote: ... Aside: If anyone is interested, we have a free video on calculating combinations (like 9C2) in your head: http://www.gmatprepnow.com/module/gmat-counting?id=789 ... Hi Brent, Thanks for this EXTREMELY useful tip! Regards, Vivek ### GMAT/MBA Expert GMAT Instructor Joined 08 Dec 2008 Posted: 13046 messages Followed by: 1253 members Upvotes: 5254 GMAT Score: 770 mevicks wrote: Brent@GMATPrepNow wrote: ... Aside: If anyone is interested, we have a free video on calculating combinations (like 9C2) in your head: http://www.gmatprepnow.com/module/gmat-counting?id=789 ... Hi Brent, Thanks for this EXTREMELY useful tip! Regards, Vivek I'm glad you like it! Cheers, Brent _________________ Brent Hanneson â€“ Creator of GMATPrepNow.com Use my video course along with Sign up for free Question of the Day emails And check out all of these free resources GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMATâ€™s FREE 60-Day Study Guide and reach your target score in 2 months! Junior | Next Rank: 30 Posts Joined 20 Nov 2013 Posted: 29 messages Upvotes: 1 Very nice tip - but one question, how would this work for other shapes, say, a triangle or parallelogram? ### GMAT/MBA Expert GMAT Instructor Joined 08 Dec 2008 Posted: 13046 messages Followed by: 1253 members Upvotes: 5254 GMAT Score: 770 For other shapes, we may be able to use pieces of the strategy. For example, if one side of a triangle were parallel to the x-axis, then the two vertices on that side of the triangle would share the same y-coordinate. Here's an example: http://www.beatthegmat.com/og-13-coordinate-geometry-t157190.html Cheers, Brent _________________ Brent Hanneson â€“ Creator of GMATPrepNow.com Use my video course along with Sign up for free Question of the Day emails And check out all of these free resources GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMATâ€™s FREE 60-Day Study Guide and reach your target score in 2 months! Junior | Next Rank: 30 Posts Joined 20 Nov 2013 Posted: 29 messages Upvotes: 1 Brent@GMATPrepNow wrote: For other shapes, we may be able to use pieces of the strategy. For example, if one side of a triangle were parallel to the x-axis, then the two vertices on that side of the triangle would share the same y-coordinate. Here's an example: http://www.beatthegmat.com/og-13-coordinate-geometry-t157190.html Cheers, Brent Thanks for that. I am not sure why we can't follow the strategy you posted for the rectangle? In other words, why not get the probably (in no particular order) of picking two points along the x axis and y axis like we did with the triangle? Is it because a rectangle flipped upside down is the same shape while a triangle flipped might look different (for example, if the coordinates of the right angle were (0,0 vs. 0,8)? ### GMAT/MBA Expert GMAT Instructor Joined 12 Sep 2012 Posted: 2635 messages Followed by: 117 members Upvotes: 625 Target GMAT Score: V51 GMAT Score: 780 Zach.J.Dragone wrote: Brent@GMATPrepNow wrote: For other shapes, we may be able to use pieces of the strategy. For example, if one side of a triangle were parallel to the x-axis, then the two vertices on that side of the triangle would share the same y-coordinate. Here's an example: http://www.beatthegmat.com/og-13-coordinate-geometry-t157190.html Cheers, Brent Thanks for that. I am not sure why we can't follow the strategy you posted for the rectangle? In other words, why not get the probably (in no particular order) of picking two points along the x axis and y axis like we did with the triangle? Is it because a rectangle flipped upside down is the same shape while a triangle flipped might look different (for example, if the coordinates of the right angle were (0,0 vs. 0,8)? Yeah, that's essentially it. I don't think Brent was dismissing your suggestion; he was (wisely) refraining from endorsing this as a universally applicable one- or two-step method. You always have to consider rotations and reflections and determine which shapes are unique - these sorts of problems get well beyond GMAT difficulty very quickly. Enroll in a Veritas Prep GMAT class completely for FREE. Wondering if a GMAT course is right for you? Attend the first class session of an actual GMAT course, either in-person or live online, and see for yourself why so many students choose to work with Veritas Prep. Find a class now! Junior | Next Rank: 30 Posts Joined 22 Jun 2016 Posted: 13 messages Upvotes: 1 Quick Question: While did we not multiply 1980 with 4! as we did in the 3 consonants and 2 vowels question(http://www.beatthegmat.com/how-many-words-t279187.html)? ### GMAT/MBA Expert GMAT Instructor Joined 08 Dec 2008 Posted: 13046 messages Followed by: 1253 members Upvotes: 5254 GMAT Score: 770 Paras_0111 wrote: Quick Question: While did we not multiply 1980 with 4! as we did in the 3 consonants and 2 vowels question(http://www.beatthegmat.com/how-many-words-t279187.html)? Here's an illustrative example to explain why. The points A(8, -2), B(11, -2), C(8, 4) and D(11, 4) create a rectangle. The points B(8, -2), D(11, -2), A(8, 4) and C(11, 4) create the same rectangle. Cheers, Brent _________________ Brent Hanneson â€“ Creator of GMATPrepNow.com Use my video course along with Sign up for free Question of the Day emails And check out all of these free resources GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMATâ€™s FREE 60-Day Study Guide and reach your target score in 2 months! Master | Next Rank: 500 Posts Joined 26 Apr 2014 Posted: 199 messages Followed by: 4 members Upvotes: 16 Test Date: 7/9/2016 GMAT Score: 780 Matt@VeritasPrep wrote: Zach.J.Dragone wrote: Brent@GMATPrepNow wrote: For other shapes, we may be able to use pieces of the strategy. For example, if one side of a triangle were parallel to the x-axis, then the two vertices on that side of the triangle would share the same y-coordinate. Here's an example: http://www.beatthegmat.com/og-13-coordinate-geometry-t157190.html Cheers, Brent Thanks for that. I am not sure why we can't follow the strategy you posted for the rectangle? In other words, why not get the probably (in no particular order) of picking two points along the x axis and y axis like we did with the triangle? Is it because a rectangle flipped upside down is the same shape while a triangle flipped might look different (for example, if the coordinates of the right angle were (0,0 vs. 0,8)? Yeah, that's essentially it. I don't think Brent was dismissing your suggestion; he was (wisely) refraining from endorsing this as a universally applicable one- or two-step method. You always have to consider rotations and reflections and determine which shapes are unique - these sorts of problems get well beyond GMAT difficulty very quickly. Very true. This is already a very difficult counting question as is. Most other geometric shapes would probably produce results way outside the scope of the GMAT. Correct me if I'm wrong, but for a triangle... It would be 99C3, and then subtract all the combinations that produce 3 collinear points, which would be extremly difficult in a grid of this size. Especially starting from a total possibility of 99x98x97/6. _________________ 800 or bust! ### GMAT/MBA Expert GMAT Instructor Joined 12 Sep 2012 Posted: 2635 messages Followed by: 117 members Upvotes: 625 Target GMAT Score: V51 GMAT Score: 780 800_or_bust wrote: Correct me if I'm wrong, but for a triangle... It would be 99C3, and then subtract all the combinations that produce 3 collinear points, which would be extremly difficult in a grid of this size. Especially starting from a total possibility of 99x98x97/6. Any time you're even using a word like concurrent or collinear, you know you're outside of the realm of the GMAT! Enroll in a Veritas Prep GMAT class completely for FREE. Wondering if a GMAT course is right for you? Attend the first class session of an actual GMAT course, either in-person or live online, and see for yourself why so many students choose to work with Veritas Prep. Find a class now! Junior | Next Rank: 30 Posts Joined 16 Jan 2017 Posted: 24 messages I solved it this way: Rectangle ABCD Stage 1: Choosing point A. The grid for this point is 9*11= 99 Stage 2: Choosing Point B. As Point B lies on the same y coordinate, we have 1 value for y and 8 possible values for x as one value of x is already taken for point A. Point B can be chosen in 8*1= 8 Stage 3: Choosing Point C. As Point C lies on the same x coordinate, we have 1 value for x and 10 possible values for y as one value of y is already taken for point A. Point C can be chosen in 1*10= 10 Stage 4: Choosing Point D. After choosing points A,B, and C the shape locks down and we have only 1 way to choose point D. So, Total ways of shaping the rectangle is 99*8*10*1= 7920 What is wrong of my approach? Thanks ### GMAT/MBA Expert GMAT Instructor Joined 08 Dec 2008 Posted: 13046 messages Followed by: 1253 members Upvotes: 5254 GMAT Score: 770 Zoser wrote: I solved it this way: Rectangle ABCD Stage 1: Choosing point A. The grid for this point is 9*11= 99 Stage 2: Choosing Point B. As Point B lies on the same y coordinate, we have 1 value for y and 8 possible values for x as one value of x is already taken for point A. Point B can be chosen in 8*1= 8 Stage 3: Choosing Point C. As Point C lies on the same x coordinate, we have 1 value for x and 10 possible values for y as one value of y is already taken for point A. Point C can be chosen in 1*10= 10 Stage 4: Choosing Point D. After choosing points A,B, and C the shape locks down and we have only 1 way to choose point D. So, Total ways of shaping the rectangle is 99*8*10*1= 7920 What is wrong of my approach? Thanks This approach allows for identical rectangles to be counted more than once. For example, in your approach, we might select (4,3) for point A, then select (10,3) for point B, then (4,-5) for point C and then (10, -5) for point D This gets counted as 1 outcome. However, if we select (10, -5) for point A, then select (4,-5) for point B, then (10,3) for point C and then (4, 3) for point D, this gets counted as a different outcome (which it isn't). Cheers, Brent _________________ Brent Hanneson â€“ Creator of GMATPrepNow.com Use my video course along with Sign up for free Question of the Day emails And check out all of these free resources GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMATâ€™s FREE 60-Day Study Guide and reach your target score in 2 months! Junior | Next Rank: 30 Posts Joined 16 Jan 2017 Posted: 24 messages Quote: This approach allows for identical rectangles to be counted more than once. For example, in your approach, we might select (4,3) for point A, then select (10,3) for point B, then (4,-5) for point C and then (10, -5) for point D This gets counted as 1 outcome. However, if we select (10, -5) for point A, then select (4,-5) for point B, then (10,3) for point C and then (4, 3) for point D, this gets counted as a different outcome (which it isn't). Cheers, Brent Thanks Brent for the prompt reply! Can you tell me why in this question you somehow followed the same approach I did that led to multiple counting? http://www.beatthegmat.com/og-13-coordinate-geometry-t157190.html • FREE GMAT Exam Know how you'd score today for$0

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