Let S be the set of points (a, b) in the coordinate plane, where each

Question

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

Archit3110 · Answer

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

chetan2u · Answer

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

Maxsparrow · Answer

I cannot comphrehend the combinations method to solve this, so I list out the possibility

Y constant: 3 lines
X constant: 3 lines
m = 1 : 3 lines (m = gradient)
m = -1: 3 lines (always get the negative gradient as an alternative)
m = 2: 2 lines
m = -2: 2 lines
m = 1/2: 2 lines
m = -1/2: 2 lines
Total: 3*4 + 2*4 = 5*4 = 20

bumpbot · Answer

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Let S be the set of points (a, b) in the coordinate plane, where each

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