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How many triangles with positive area have all their vertices at point

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Bunuel
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How many triangles with positive area have all their vertices at point [#permalink] New post   10 May 2019, 01:09
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25% (02:11) correct 75% (02:38) wrong based on 93 sessions
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How many triangles with positive area have all their vertices at points (i,j) in the coordinate plane, where i and j are integers between 1 and 5, inclusive?

(A) 2128

(B) 2148

(C) 2160

(D) 2200

(E) 2300
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B
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Sneha333
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Re: How many triangles with positive area have all their vertices at point [#permalink] New post   14 May 2019, 11:34
C should be the answer.

Area of Triangle is always +ve. Any three non co linear points here will make a positive area.
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SPatel1992
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Re: How many triangles with positive area have all their vertices at point [#permalink] New post   14 May 2019, 20:33
Hi! Can someone explain the solution?
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globaldesi
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Re: How many triangles with positive area have all their vertices at point [#permalink] New post   14 May 2019, 21:54
Co-ordinates can be (1,1), (1,2),(1,3),(1,4) and (1,5)
similary (2,1)...(5,5)
total of 25
1. select any three from these 25
that is \(25C_3\)
but we cant select all three from (1,1), (1,2),(1,3),(1,4) and (1,5) otherwise it will be a linear line
so \(25C_3\)-5*\(5C_3\)
i am stuck after it

chetan2u can you please help what else am i missing?
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nick1816
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Re: How many triangles with positive area have all their vertices at point [#permalink] New post   14 May 2019, 22:18
You also have to deduct the cases when 3 points lie on the diagonally straight lines. I'm getting 2160, but OA is B. :(

globaldesi wrote:
Co-ordinates can be (1,1), (1,2),(1,3),(1,4) and (1,5)
similary (2,1)...(5,5)
total of 25
1. select any three from these 25
that is \(25C_3\)
but we cant select all three from (1,1), (1,2),(1,3),(1,4) and (1,5) otherwise it will be a linear line
so \(25C_3\)-5*\(5C_3\)
i am stuck after it

chetan2u can you please help what else am i missing?
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globaldesi
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Re: How many triangles with positive area have all their vertices at point [#permalink] New post   14 May 2019, 22:39
nick1816 wrote:
You also have to deduct the cases when 3 points lie on the diagonally straight lines. I'm getting 2160, but OA is B. :(

globaldesi wrote:
Co-ordinates can be (1,1), (1,2),(1,3),(1,4) and (1,5)
similary (2,1)...(5,5)
total of 25
1. select any three from these 25
that is \(25C_3\)
but we cant select all three from (1,1), (1,2),(1,3),(1,4) and (1,5) otherwise it will be a linear line
so \(25C_3\)-5*\(5C_3\)
i am stuck after it

chetan2u can you please help what else am i missing?


Agree.
We are missing 12 points.

Bunuel what other points did we miss? Or is there another way to solve it

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Bunuel
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Re: How many triangles with positive area have all their vertices at point [#permalink] New post   14 May 2019, 22:54
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Bunuel wrote:
How many triangles with positive area have all their vertices at points (i,j) in the coordinate plane, where i and j are integers between 1 and 5, inclusive?

(A) 2128

(B) 2148

(C) 2160

(D) 2200

(E) 2300


Official Explanation:



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Attachment:
2019-05-15_0953.png
2019-05-15_0953.png [ 95.15 KiB | Viewed 3433 times ]
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Sneha333
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How many triangles with positive area have all their vertices at point [#permalink] New post   15 May 2019, 02:02
Please explain the last 3 points w.r.t to slope mentioned in the solution.
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nick1816
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Re: How many triangles with positive area have all their vertices at point [#permalink] New post   15 May 2019, 02:25
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Top one is depicting lines of slope 2 and -2
Bottom one is depicting lines of slope -1/2 and 1/2

Sneha333 wrote:
Please explain the last 3 points w.r.t to slope mentioned in the solution.

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Sneha333
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Re: How many triangles with positive area have all their vertices at point [#permalink] New post   15 May 2019, 02:38
nick1816 wrote:
Top one is depicting lines of slope 2 and -2
Bottom one is depicting lines of slope -1/2 and 1/2

Sneha333 wrote:
Please explain the last 3 points w.r.t to slope mentioned in the solution.


Thanks Nick for the explanation.
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Neha2050
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Re: How many triangles with positive area have all their vertices at point [#permalink] New post   25 Jun 2019, 08:20
Bunuel wrote:
Bunuel wrote:
How many triangles with positive area have all their vertices at points (i,j) in the coordinate plane, where i and j are integers between 1 and 5, inclusive?

(A) 2128

(B) 2148

(C) 2160

(D) 2200

(E) 2300


Official Explanation:



Show Spoiler
Attachment:
2019-05-15_0953.png


i am unable to understand the point from collinear point calculation
can u elaborate pls
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Re: How many triangles with positive area have all their vertices at point [#permalink] New post   05 Jul 2019, 21:37
Bunuel wrote:
Bunuel wrote:
How many triangles with positive area have all their vertices at points (i,j) in the coordinate plane, where i and j are integers between 1 and 5, inclusive?

(A) 2128

(B) 2148

(C) 2160

(D) 2200

(E) 2300


Official Explanation:



Show Spoiler
Attachment:
2019-05-15_0953.png


Hi the solution is spot on. I missed a few cases ie. the ones with slopes +- 1/2 and +- 2. Can we expect questions like these on the GMAT? Are they not tad bit more time consuming and calculation intensive?
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Re: How many triangles with positive area have all their vertices at point [#permalink]
05 Jul 2019, 21:37
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