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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

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

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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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.

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

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

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

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26 Apr 2015, 14:17

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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

Answer is n(n-1)/2 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

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