Vincen wrote:A right triangle ABC has to be constructed in the xy-plane so that the right angle is at A and AB is parallel to x axis. The coordinates of A, B and C are integers to satisfy the inequalities -1 ≤ x ≤ 7 and 1 ≤ y ≤ 7. The numbers of different triangles that can be constructed with these properties are?
A. 63
B. 336
C. 567
D. 3024
E. 5040
Take the task of building triangles and break it into stages.
Stage 1: Select any point where the right angle will be (point A).
The point can be selected from a 9x7 grid. So, there 63 points to choose from.
This means that stage 1 can be completed in
63 ways.
Stage 2: Select a point that is on the same horizontal line as the first point (point A). This point will be point B.
The 2 legs of the right triangle are parallel to the x- and y-axes.
The first point we select (in stage 1) dictates the y-coordinate of point A.
In how many ways can we select the x-coordinate of point B?
Well, we can choose any of the 9 coordinates from -1 to 7 inclusive EXCEPT for the x-coordinate we chose for point A (in stage 1).
So, there are 8 coordinates to choose from.
This means that stage 2 can be completed in
8 ways.
Stage 3: Select a point that is on the same vertical line as the first point. This point will be point C.
The 2 legs of the right triangle are parallel to the x- and y-axes.
The first point we select (in stage 1) dictates the x-coordinate of point A.
In how many ways can we select the y-coordinate of point C?
Well, we can choose any of the 1 integral coordinates from 1 to 7 inclusive EXCEPT for the y-coordinate we chose for point A (in stage 1).
So, there are 6 coordinates to choose from.
This means that stage 3 can be completed in
6 ways.
So, by the Fundamental Counting Principle (FCP), the total number of triangles = (
63)(
8)(
6) = [spoiler]3024 = D[/spoiler]
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Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. For more information about the FCP, watch our free video:
https://www.gmatprepnow.com/module/gmat-counting?id=775
Then you can try solving the following questions:
EASY
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https://www.beatthegmat.com/what-should- ... 67256.html
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https://www.beatthegmat.com/counting-pro ... 44302.html
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https://www.beatthegmat.com/picking-a-5- ... 73110.html
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https://www.beatthegmat.com/permutation- ... 57412.html
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https://www.beatthegmat.com/simple-one-t270061.html
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https://www.beatthegmat.com/mouse-pellets-t274303.html
MEDIUM
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https://www.beatthegmat.com/combinatoric ... 73194.html
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https://www.beatthegmat.com/arabian-hors ... 50703.html
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https://www.beatthegmat.com/sub-sets-pro ... 73337.html
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https://www.beatthegmat.com/combinatoric ... 73180.html
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https://www.beatthegmat.com/digits-numbers-t270127.html
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https://www.beatthegmat.com/doubt-on-sep ... 71047.html
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https://www.beatthegmat.com/combinatoric ... 67079.html
DIFFICULT
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https://www.beatthegmat.com/wonderful-p- ... 71001.html
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https://www.beatthegmat.com/ps-counting-t273659.html
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https://www.beatthegmat.com/permutation- ... 73915.html
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https://www.beatthegmat.com/please-solve ... 71499.html
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https://www.beatthegmat.com/no-two-ladie ... 75661.html
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https://www.beatthegmat.com/laniera-s-co ... 15764.html
Cheers,
Brent
