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What is the maximum number of triangles that can be formed by plotting

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What is the maximum number of triangles that can be formed by plotting  [#permalink]

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New post Updated on: 01 Apr 2020, 06:52
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What is the maximum number of triangles that can be formed by plotting 19 points on three parallel lines?

A. 300
B. 416
C. 540
D. 642
E. 894

Originally posted by Dillesh4096 on 01 Apr 2020, 06:25.
Last edited by Dillesh4096 on 01 Apr 2020, 06:52, edited 1 time in total.
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Re: What is the maximum number of triangles that can be formed by plotting  [#permalink]

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New post 01 Apr 2020, 06:37
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Dillesh4096 wrote:
What is the maximum number of triangles that can be formed by plotting 19 points on three parallel lines?

A. 300
B. 416
C. 540
D. 642
E. 756


Let's take the points equally distributed

i.e. 6 points on Line 1, 6 points on Line 2, and 7 points on Line 3

Total Triangles = 19C3 - (6C3+6C3+7C3) = 969 - (20+20+35) = 894
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Re: What is the maximum number of triangles that can be formed by plotting  [#permalink]

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New post 01 Apr 2020, 08:07
Dillesh4096 wrote:
What is the maximum number of triangles that can be formed by plotting 19 points on three parallel lines?

A. 300
B. 416
C. 540
D. 642
E. 894


total possible ∆ ; 19c3
and for these 19 points to be plotted on 3 ll lines ; the points are to be plotted on each line ( 7,6,6)
so we get 7c3+6c3+6c3
total possible ∆
19c3-(7c3+6c3+6c3) ;
969- ( 35+20+20)
=> 894
IMO E
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Re: What is the maximum number of triangles that can be formed by plotting  [#permalink]

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New post 01 Apr 2020, 14:02
If there are p,q, r points on three parallel lines ,the number of triangles which can be formed with vertices at their points is
(p+q+r)C3− (pC3 + qC3 + rC3)
now, to get max no of triangles (pC3 + qC3 + rC3) will be minimum.
Here, p+q+r = 19
now, 19 can be distributed in p,q,r in several ways, such as (10,5,4) , (5,6,8), (6,6,7) , ... and so on. from max- min concept when all the values are distributed equally , the resultant value will give either max or min. so here the logical combination will be (6,6,7)

so, the max required triangles can be formed 19C3- 6C3-6C3-7C3 = 969-20-20-35 = 894.
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Re: What is the maximum number of triangles that can be formed by plotting  [#permalink]

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New post 02 Apr 2020, 04:29
another approach, please correct if wrong..

Question summary: 19 points plotted 3 parallel lines, how many triangles can be made?

=> 19 points and to be distributed in 3 groups.
=> 19C3 = 969

Now for 19 points in 3 lines, each line A, B, C should have max 6,6,7 points for equal distribution since we need maximum lines available for cross intersections (triangle formation)

Therefore number of ways 3 points can be selected from each line A, B, C are 6C3 , 6C3, 7C3
Sum = 6C3 + 6C3 + 7C3 = 75 (this defys triangle property since no 3 points can be on same line)

These ways should be subtracted from the total to get number of triangles available = 969 - 75 = 894
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Re: What is the maximum number of triangles that can be formed by plotting   [#permalink] 02 Apr 2020, 04:29

What is the maximum number of triangles that can be formed by plotting

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