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# If 25 lines are drawn in a plane such that no two of them

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Senior Manager
Joined: 13 Dec 2006
Posts: 301
Location: Indonesia
If 25 lines are drawn in a plane such that no two of them  [#permalink]

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06 Dec 2007, 05:04
3
48
00:00

Difficulty:

55% (hard)

Question Stats:

58% (01:42) correct 42% (01:41) wrong based on 477 sessions

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If 25 lines are drawn in a plane such that no two of them are parallel and no three are concurrent, then in how many points do they intersect?

A. 2300
B. 600
C. 250
D. 300
E. none of these
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 10485
Location: Pune, India
Re: If 25 lines are drawn in a plane such that no two of them  [#permalink]

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24 Nov 2013, 20:42
13
5
Amardeep Sharma wrote:
If 25 lines are drawn in a plane such that no two of them are parallel and no three are concurrent, then in how many points do they intersect?

A 2300
B 600
C 250
D 300
E none of these

Please explaing for me to understand the concept

Amar

Responding to a pm:

We need to draw lines such that they are not parallel. Why is 'not parallel' important? Any two distinct lines drawn on the xy axis will either be parallel or will intersect in exactly one point. Lines can be extended infinitely on both ends so somewhere they will intersect with each other if they are not parallel. Since any given two lines are not parallel, we can say that they must intersect at exactly one point. So every pair of two lines will intersect at exactly one point. We are also given that no three lines are concurrent. This means that no three lines intersect at the same point. So every pair of two lines we select will have a unique point of intersection which they will not share with any third line. So how many such unique points of intersection do we get? That depends on how many pairs of 2 lines can we select from the 25 lines?
We can select 2 lines from 25 lines in 25C2 ways i.e. 300 ways. Each one of these pairs will give us one unique point of intersection so we will get 300 points of intersection.

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06 Dec 2007, 05:19
8
5
D.

let try draw lines one by one.

1st line - 0 points
2nd line - new 1 point
3td line - new 2 points + old 1 point
4th line - new 3 points + old 2+1 points
5th line - new 4 points + old 3+2+1 points
n-th line - (n-1) points + (n-2) .... 3+2+1

therefore, S=1+2+3...(n-1)

S=(n-1)n/2=24*25/2=300
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Joined: 22 Nov 2007
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06 Dec 2007, 05:27
It is very hard. I can only say that the answer must not be E, because "no two of them are parallel", so they must meet in some point.
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06 Dec 2007, 05:30
Can you explain me your reasoning step by step very slowly?
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06 Dec 2007, 05:37
3
1
marcodonzelli wrote:
Can you explain me your reasoning step by step very slowly?

No. less than 2 min.

Another way, even easier and faster:

one line has 24 intersections. We have 25 line. Therefore the number of intersection points is 24*25/2 (2- because we count twice the same point)
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07 Dec 2007, 11:08
The question should state that the lines are of infinite length otherwise the answer could be anything.
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07 Dec 2007, 11:49
2
1
alexperi wrote:
The question should state that the lines are of infinite length otherwise the answer could be anything.

see difference between "line" and "line segment" in geometry
http://www.mathleague.com/help/geometry/basicterms.htm or
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08 Dec 2007, 03:39
3
1
Amardeep Sharma wrote:
If 25 lines are drawn in a plane such that no two of them are parallel and no three are concurrent, then in how many points do they intersect?

A 2300
B 600
C 250
D 300
E none of these

Please explaing for me to understand the concept

Amar

I got D:
since any 2 lines has 1 intersect with each other, we need to find the number of ways to choose 2 out of 25: 25C2=25*24/2=300.
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25 Aug 2008, 12:26
1
Amardeep Sharma wrote:
If 25 lines are drawn in a plane such that no two of them are parallel and no three are concurrent, then in how many points do they intersect?

A 2300
B 600
C 250
D 300
E none of these

Please explaing for me to understand the concept

Amar

2 lines form 1 intersect points =2C2
3 lines form 3 intersect points= 3C2
25 llines form 25C2 intersect points = 25*12=300.

D
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Joined: 27 Oct 2008
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27 Sep 2009, 20:12
3
1
If 25 lines are drawn in a plane such that no two of them are parallel and no three are concurrent, then in how many points do they intersect?

A 2300
B 600
C 250
D 300
E none of these

Soln: Since no three are concurrent, hence any point that is formed by two different lines are distinct.
The first line intersects each of the other 24 lines at 24 points. => statement 1
The second line intersects each of the other 23 lines at 23 points. The point with first line has already been counted in the statement no.1.
The third line intersects each of the other 22 lines at 22 points
and so on.

Thus total number of points is
= 24 + 23 + 22 + ... + 1
= 24 * 25/2
= 300

Ans is D
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16 Feb 2010, 07:41
1
Amardeep Sharma wrote:
If 25 lines are drawn in a plane such that no two of them are parallel and no three are concurrent, then in how many points do they intersect?

A 2300
B 600
C 250
D 300
E none of these

Please explaing for me to understand the concept

Amar

No three lines are concurrent and no two lines are parallel gives us the info that every line intersects the other and no intersection point is common. Hence no of intersection points = 25c2 = 300 = D
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Re: If 25 lines are drawn in a plane such that no two of them  [#permalink]

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12 Dec 2013, 22:28
The answer is easier than it seems to be
25C2=300
As any two lines have exactly 1 intersection point (just draw a few non-parallel lines), we simply need to find in how many ways we can chose 2 lines out of 25
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Re: If 25 lines are drawn in a plane such that no two of them  [#permalink]

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13 Dec 2013, 08:57
I think Walker made it solveable. Thanks
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Joined: 28 Jan 2013
Posts: 26
Re: If 25 lines are drawn in a plane such that no two of them  [#permalink]

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26 Feb 2014, 16:26
Amardeep Sharma wrote:
If 25 lines are drawn in a plane such that no two of them are parallel and no three are concurrent, then in how many points do they intersect?

A. 2300
B. 600
C. 250
D. 300
E. none of these

No 3 lines intersect at one point.. and none of them are parallel...
point is created when 2 lines intersect... how many ways can you select 2 out of 25 = 25C2=300
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Re: If 25 lines are drawn in a plane such that no two of them  [#permalink]

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04 Jul 2015, 09:56
Amardeep Sharma wrote:
If 25 lines are drawn in a plane such that no two of them are parallel and no three are concurrent, then in how many points do they intersect?

A. 2300
B. 600
C. 250
D. 300
E. none of these

Every pair of two lines causes one Point of Intersection

So the No. of ways in which pairs of two lines out of 25 lines can be selected in different ways = 25C2 = 300

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Re: If 25 lines are drawn in a plane such that no two of them  [#permalink]

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13 Aug 2016, 09:39
Amardeep Sharma wrote:
If 25 lines are drawn in a plane such that no two of them are parallel and no three are concurrent, then in how many points do they intersect?

A. 2300
B. 600
C. 250
D. 300
E. none of these

Just remember a simple thing, same concept applies to handshakes, no. of matches in tournament. nC2 is the answer.
Hence (25X24)/2=300 D
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Re: If 25 lines are drawn in a plane such that no two of them  [#permalink]

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18 Aug 2016, 20:23
Amardeep Sharma wrote:
If 25 lines are drawn in a plane such that no two of them are parallel and no three are concurrent, then in how many points do they intersect?

A. 2300
B. 600
C. 250
D. 300
E. none of these

The keywords are "no two of them are parallel" and "no three are concurrent". The former meaning, given any two lines, they intersect at only point and the latter meaning at any intersection points, it's only 2 lines that are intersecting and not more than that. This is to ensure that every point of intersection is only between 2 lines.

So 1 pair of lines (2 lines) intersect at 1 point
3 lines intersect at 3 point , i.e, from 3 choose as a pair(2) , i.e 3C2 =3
4 lines intersect at, from 4 choose as a pair(2) = 4C2 = 6

So from 25 lines, choose in pairs = 25 C 2 = 25*24/2 = 300

+1 for kudos
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Re: If 25 lines are drawn in a plane such that no two of them  [#permalink]

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18 Aug 2016, 20:34
Is this really a 700-level question?
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Re: If 25 lines are drawn in a plane such that no two of them  [#permalink]

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17 Apr 2017, 17:09
Amardeep Sharma wrote:
If 25 lines are drawn in a plane such that no two of them are parallel and no three are concurrent, then in how many points do they intersect?

A. 2300
B. 600
C. 250
D. 300
E. none of these

GMAT Experts
How would be the approach if 3 lines for instance would be parallel?

Tks
Re: If 25 lines are drawn in a plane such that no two of them   [#permalink] 17 Apr 2017, 17:09

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