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S is a set of points in the plane. How many distinct triangles can be

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Re: S is a set of points in the plane. How many distinct triangles can be  [#permalink]

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New post 23 Jul 2018, 22:23
ENEM wrote:
what does 'no three of the points in S are collinear' mean?
Also is this really a sub 600 level question? Shouldn't it be at least 600-700 level?


"Collinear" means lying on the same straight line.

3 points are collinear when you can draw a line that passes through all 3 of them. If this happens, the three points cannot make a triangle.
Note that you can always draw a line with any two points by just joining the 2 points.
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Re: S is a set of points in the plane. How many distinct triangles can be  [#permalink]

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New post 23 Jul 2018, 22:33
KarishmaB

Hi,

no I meant what does it mean 'no three points' every three points in set S? What does that mean?
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Re: S is a set of points in the plane. How many distinct triangles can be  [#permalink]

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New post 24 Jul 2018, 01:38
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ENEM wrote:
KarishmaB

Hi,

no I meant what does it mean 'no three points' every three points in set S? What does that mean?


It means no matter which 3 points you pick up, they will not be collinear.
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Re: S is a set of points in the plane. How many distinct triangles can be  [#permalink]

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New post 25 Jul 2018, 04:42
walker wrote:
C

1. we cannot create triangles for 5 points of a line but can do that for points that are not collinear. insuff.
2. we don't know the number of points insuff.

1&2 \(C^5_3=\frac{5*4*3*2}{3*2*2}=10\)



Can you please write formula to calculate the number of triangles if only 3 points are coliner
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Re: S is a set of points in the plane. How many distinct triangles can be  [#permalink]

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New post 25 Jul 2018, 10:49
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kablayi wrote:
walker wrote:
C

1. we cannot create triangles for 5 points of a line but can do that for points that are not collinear. insuff.
2. we don't know the number of points insuff.

1&2 \(C^5_3=\frac{5*4*3*2}{3*2*2}=10\)



Can you please write formula to calculate the number of triangles if only 3 points are coliner


Hello

Ok, here's a general formula for calculating possible number of triangles.

If there are N points in a plane, and no three of them are collinear, then the number of possible triangles = N(C)3. (number of possible selections of 3 objects from N objects)

If there are N points in a plane, and X of them are collinear, then the number of possible triangles = N(C)3 - X(C)3. (subtracting number of possible selections of 3 objects from X FROM number of possible seelctions of 3 objects from N).

Now to answer your question: IF we were given that there are 5 points in a plane, out of which 3 are collinear, then number of possible triangles would have been = 5C3 - 3C3 = 10 - 1 = 9. Thus 9 triangles would have been possible.
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Re: S is a set of points in the plane. How many distinct triangles can be  [#permalink]

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New post 12 Aug 2018, 04:20
KarishmaB wrote:
ENEM wrote:
KarishmaB

Hi,

no I meant what does it mean 'no three points' every three points in set S? What does that mean?


It means no matter which 3 points you pick up, they will not be collinear.


Dear Karishma / all,

I'm struggling to understand one thing: as a plane is not always a polygon, can the attached image be considered as a 5-point plane which satisfies the 2 conditions per the question ? If so, then wouldn't the answer suppose to be E as we cant confirm the number of triangles then ?

Many thanks in Advance!
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5 points plane.png
5 points plane.png [ 31.57 KiB | Viewed 667 times ]

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Re: S is a set of points in the plane. How many distinct triangles can be  [#permalink]

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New post 11 Sep 2018, 18:42
Bunuel wrote:
ranaazad wrote:
GGUY wrote:
S is a set of points in the plane. How many distinct triangles can be drawn that have three of the points in S as vertices?

(1) The number of distinct points in S is 5.
(2) No three of the points in S are collinear.


l got A. What does collinear mean? :oops:


Collinear points are those that lie on the same straight line. BTW the correct answer is C, not A. Check the solutions above and ask if anything remains unclear.


To be fair, I didn't even understand the question nor the answers given.
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Re: S is a set of points in the plane. How many distinct triangles can be  [#permalink]

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New post 15 Dec 2018, 06:20
Given: “S” has set of points in the plane.

To find: How many number of distinct triangles can be drawn from set S?

Inference: We need three distinct points to form a triangle which is not in a straight line, which means they shouldn’t be collinear.

Analysis of statement 1: The number of distinct points in S is 5.
As we think that this statement provides us with the information about how many points are there in set S, to form triangle we need 3 points, therefore the number of triangles formed will be \(= 5C2 = 10.\)
I can say that this statement is a "trap" where we tend to make mistake by choosing the answer statement 1 is sufficient.
Here we can have two cases:
Case1: If all the 5 points are on the same straight line, then we cannot form the triangle.
Case 2: If all the 5 points are not collinear in nature then we can form the triangles.
So we have to be very careful regarding the trap.
Hence statement 1 is not sufficient to answer. We can eliminate options A and D.

Analysis of statement 2: No three of the points in S are collinear.
This statement provides us with information about three points in set S are collinear in nature, but we do not have the information about the total number of points. The number triangles are dependent on the total number of points.
Hence statement 2 is not sufficient to answer. We can eliminate option B.

Combining the statements 1 and 2 together; we get:
From statement 1: The number of points in set S is “5”
From statement 2: Three points in set S are collinear.
So, we can select 3 points from 5 points to form a triangle.
Number of triangles = \(5C3\)= \(\frac{(5 ×4 ×3)}{(3 ×2 ×1)}\) = 10.
Therefore combining the statements 1 and 2 we get the answer.

So, the correct answer option is “C”.

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Re: S is a set of points in the plane. How many distinct triangles can be  [#permalink]

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New post 28 Dec 2018, 15:01
I agree with Rachit Shah in the posts above, what if we get impossible cases from the points like 5,6,13 where triangle is not possible. What if by using the formula 5C3 we are including an impossible case in our count?
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Re: S is a set of points in the plane. How many distinct triangles can be   [#permalink] 28 Dec 2018, 15:01

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