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

Question

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

KarishmaB · Accepted Answer

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.

Answer (D)

walker · Answer

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

marcodonzelli · Answer

Can you explain me your reasoning step by step very slowly?

walker · Answer

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)

alexperi · Answer

The question should state that the lines are of infinite length otherwise the answer could be anything.

walker · Answer

see difference between "line" and "line segment" in geometry https://www.mathleague.com/help/geometry/basicterms.htm or https://math.about.com/library/weekly/aa031503a.htm

elgo · Answer

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.

srivas · Answer

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

GMATinsight · Answer

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 = 25 C 2 = 300

Answer: Option D

Fdambro294 · Answer

Rule: unless 2 lines are parallel, the 2 lines drawn in a plane will eventually always intersect at one point

Since none of the 25 lines are parallel to each other, all 25 lines will intersect with one another at one point in the plane

Line 1 + Line 2 ———-> will create 1 intersection point

When line 3 is drawn, it will intersect with line 1 and line 2———-> +2 more intersection points

When line 4 is drawn, it will intersect with line 1 and line 2 and line 3 ———> +3 more intersection points

The pattern will continue until:

When line 25 is drawn ———> +24 more intersection points

We just need to sum the consecutive positive integers from 1 through 24:

= (24) (1 + 24) / 2

= 12 * 25

300

Posted from my mobile device

bumpbot · Answer

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards