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Math Expert V
Joined: 02 Sep 2009
Posts: 58428
Let S be the set of points (a, b) in the coordinate plane, where each  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 25% (02:13) correct 75% (02:00) wrong based on 48 sessions

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Let S be the set of points (a, b) in the coordinate plane, where each of a and b may be -1, 0, or 1. How many distinct lines pass through at least two members of S?

(A) 8
(B) 20
(C) 24
(D) 27
(E) 36

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Joined: 18 Aug 2017
Posts: 5031
Location: India
Concentration: Sustainability, Marketing
GPA: 4
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Let S be the set of points (a, b) in the coordinate plane, where each  [#permalink]

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Bunuel wrote:
Let S be the set of points (a, b) in the coordinate plane, where each of a and b may be -1, 0, or 1. How many distinct lines pass through at least two members of S?

(A) 8
(B) 20
(C) 24
(D) 27
(E) 36

total pairs of points which can be formed is 9
and line passing through 2 such points 9c2 = 36

we have overcounted all of the lines which pass through three points. In fact, each line which passes through three points will have been counted 3c2 = 3 times
3 horizontal, 3 vertical, and 2 diagonal lines, so the answer is 36 - 2(3+3+2) = 20
IMO B
Intern  B
Joined: 09 Apr 2018
Posts: 13
Re: Let S be the set of points (a, b) in the coordinate plane, where each  [#permalink]

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Archit3110 wrote:
Bunuel wrote:
Let S be the set of points (a, b) in the coordinate plane, where each of a and b may be -1, 0, or 1. How many distinct lines pass through at least two members of S?

(A) 8
(B) 20
(C) 24
(D) 27
(E) 36

total pairs of points which can be formed is 9
and line passing through 2 such points 9c2 = 36

we have overcounted all of the lines which pass through three points. In fact, each line which passes through three points will have been counted 3c2 = 3 times
3 horizontal, 3 vertical, and 2 diagonal lines, so the answer is 36 - 2(3+3+2) = 20
IMO B

Can you please elaborate the process for counting lines passing through 3 points?

Posted from my mobile device
Manager  B
Joined: 11 Aug 2017
Posts: 59
Re: Let S be the set of points (a, b) in the coordinate plane, where each  [#permalink]

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Archit3110 wrote:
Bunuel wrote:
Let S be the set of points (a, b) in the coordinate plane, where each of a and b may be -1, 0, or 1. How many distinct lines pass through at least two members of S?

(A) 8
(B) 20
(C) 24
(D) 27
(E) 36

total pairs of points which can be formed is 9
and line passing through 2 such points 9c2 = 36

we have overcounted all of the lines which pass through three points. In fact, each line which passes through three points will have been counted 3c2 = 3 times
3 horizontal, 3 vertical, and 2 diagonal lines, so the answer is 36 - 2(3+3+2) = 20
IMO B

Could you please elaborate 2nd part of the explanation.. It's a bouncer for me.
Manager  B
Joined: 17 Jun 2018
Posts: 52
Schools: IMD '20
GPA: 2.84
WE: Engineering (Real Estate)
Re: Let S be the set of points (a, b) in the coordinate plane, where each  [#permalink]

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chetan2u please can you explain how to crack this one ?
Math Expert V
Joined: 02 Aug 2009
Posts: 8006
Re: Let S be the set of points (a, b) in the coordinate plane, where each  [#permalink]

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Bunuel wrote:
Let S be the set of points (a, b) in the coordinate plane, where each of a and b may be -1, 0, or 1. How many distinct lines pass through at least two members of S?

(A) 8
(B) 20
(C) 24
(D) 27
(E) 36

The points should tell you that you are looking at a square with coordinates (-1,-1), (-1,1), (1,1) and (1,-1). The centre of the square and midpoint of each side are other points..
So, in total we have 9 points. But, the centre will always be part of 3 points.

So we are left with 8 points, and we have to choose 2 out of it, so 8C2=8*7/2=28.
Now, the sides of the square consists of 3 points. We have counted each side thrice, so 4*2 are repeated. Thus, 28-8=20.
For example.. side containing points A(-1,-1), B(-1,0) and C(-1,1).. The line is only one but we have calculated it thrice in our calculations. AB, BC and AC, whereas it is just on eline AC.

B
Attachments coordi.png [ 55.58 KiB | Viewed 391 times ]

_________________ Re: Let S be the set of points (a, b) in the coordinate plane, where each   [#permalink] 25 Mar 2019, 21:07
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