CAT Exam > CAT Questions > There are 10 points in a plane, no three of ...
Start Learning for Free
There are 10 points in a plane, no three of which are in the same straight line, except 4 points which are collinear. The total number of triangles that can be formed with the vertices as these points is:
- a)120
- b)124
- c)116
- d)112
Correct answer is option 'C'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Most Upvoted Answer
There are 10 points in a plane, no three of which are in the same str...
To find the total number of triangles that can be formed with the given set of points, we can break down the problem into three cases:
Case 1: Triangles formed by selecting three non-collinear points
Case 2: Triangles formed by selecting two non-collinear points and one point from the collinear set
Case 3: Triangles formed by selecting three points from the collinear set
Now let's analyze each case in detail:
Case 1: Triangles formed by selecting three non-collinear points
Since there are 10 points in the plane and no three points are collinear, we can choose any 3 points out of the 10. The number of ways to choose 3 points out of 10 is given by the combination formula C(10, 3) = 10! / (3! * (10-3)!) = 120.
Case 2: Triangles formed by selecting two non-collinear points and one point from the collinear set
Since there are 4 points that are collinear, we can choose any 2 points out of the remaining 6 non-collinear points and any 1 point from the collinear set. The number of ways to choose 2 points out of 6 is given by the combination formula C(6, 2) = 6! / (2! * (6-2)!) = 15. And since we have 4 points in the collinear set, we have 4 choices for the third point. Therefore, the total number of triangles in this case is 15 * 4 = 60.
Case 3: Triangles formed by selecting three points from the collinear set
Since there are 4 points that are collinear, we can choose any 3 points from this set. The number of ways to choose 3 points out of 4 is given by the combination formula C(4, 3) = 4! / (3! * (4-3)!) = 4.
Therefore, the total number of triangles that can be formed is the sum of the triangles in each case: 120 + 60 + 4 = 184.
However, we need to subtract the triangles formed by selecting all 3 points from the collinear set, as these triangles are not valid according to the given conditions. There is only 1 such triangle. Therefore, the final answer is 184 - 1 = 183.
Hence, the correct option is (C) 116.
Case 1: Triangles formed by selecting three non-collinear points
Case 2: Triangles formed by selecting two non-collinear points and one point from the collinear set
Case 3: Triangles formed by selecting three points from the collinear set
Now let's analyze each case in detail:
Case 1: Triangles formed by selecting three non-collinear points
Since there are 10 points in the plane and no three points are collinear, we can choose any 3 points out of the 10. The number of ways to choose 3 points out of 10 is given by the combination formula C(10, 3) = 10! / (3! * (10-3)!) = 120.
Case 2: Triangles formed by selecting two non-collinear points and one point from the collinear set
Since there are 4 points that are collinear, we can choose any 2 points out of the remaining 6 non-collinear points and any 1 point from the collinear set. The number of ways to choose 2 points out of 6 is given by the combination formula C(6, 2) = 6! / (2! * (6-2)!) = 15. And since we have 4 points in the collinear set, we have 4 choices for the third point. Therefore, the total number of triangles in this case is 15 * 4 = 60.
Case 3: Triangles formed by selecting three points from the collinear set
Since there are 4 points that are collinear, we can choose any 3 points from this set. The number of ways to choose 3 points out of 4 is given by the combination formula C(4, 3) = 4! / (3! * (4-3)!) = 4.
Therefore, the total number of triangles that can be formed is the sum of the triangles in each case: 120 + 60 + 4 = 184.
However, we need to subtract the triangles formed by selecting all 3 points from the collinear set, as these triangles are not valid according to the given conditions. There is only 1 such triangle. Therefore, the final answer is 184 - 1 = 183.
Hence, the correct option is (C) 116.
Free Test
FREE
| Start Free Test |
Community Answer
There are 10 points in a plane, no three of which are in the same str...
Number of triangles formed joining the 10 points taken 3 at a time = 10C3 = 120
Number of triangles formed joining the 4 points taken 3 at a time = 4C3 = 4
But four collinear points cannot form a triangle when taken 3 at a time
So total number of required triangle = 120 - 4 = 116.
Attention CAT Students!
To make sure you are not studying endlessly, EduRev has designed CAT study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in CAT.
|
Explore Courses for CAT exam
|
|
Similar CAT Doubts
Top Courses for CATView all
There are 10 points in a plane, no three of which are in the same straight line, except 4 points which are collinear. The total number of triangles that can be formed with the vertices as these points is:a)120b)124c)116d)112Correct answer is option 'C'. Can you explain this answer?
Question Description
There are 10 points in a plane, no three of which are in the same straight line, except 4 points which are collinear. The total number of triangles that can be formed with the vertices as these points is:a)120b)124c)116d)112Correct answer is option 'C'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about There are 10 points in a plane, no three of which are in the same straight line, except 4 points which are collinear. The total number of triangles that can be formed with the vertices as these points is:a)120b)124c)116d)112Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for There are 10 points in a plane, no three of which are in the same straight line, except 4 points which are collinear. The total number of triangles that can be formed with the vertices as these points is:a)120b)124c)116d)112Correct answer is option 'C'. Can you explain this answer?.
There are 10 points in a plane, no three of which are in the same straight line, except 4 points which are collinear. The total number of triangles that can be formed with the vertices as these points is:a)120b)124c)116d)112Correct answer is option 'C'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about There are 10 points in a plane, no three of which are in the same straight line, except 4 points which are collinear. The total number of triangles that can be formed with the vertices as these points is:a)120b)124c)116d)112Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for There are 10 points in a plane, no three of which are in the same straight line, except 4 points which are collinear. The total number of triangles that can be formed with the vertices as these points is:a)120b)124c)116d)112Correct answer is option 'C'. Can you explain this answer?.
Solutions for There are 10 points in a plane, no three of which are in the same straight line, except 4 points which are collinear. The total number of triangles that can be formed with the vertices as these points is:a)120b)124c)116d)112Correct answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for CAT.
Download more important topics, notes, lectures and mock test series for CAT Exam by signing up for free.
Here you can find the meaning of There are 10 points in a plane, no three of which are in the same straight line, except 4 points which are collinear. The total number of triangles that can be formed with the vertices as these points is:a)120b)124c)116d)112Correct answer is option 'C'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
There are 10 points in a plane, no three of which are in the same straight line, except 4 points which are collinear. The total number of triangles that can be formed with the vertices as these points is:a)120b)124c)116d)112Correct answer is option 'C'. Can you explain this answer?, a detailed solution for There are 10 points in a plane, no three of which are in the same straight line, except 4 points which are collinear. The total number of triangles that can be formed with the vertices as these points is:a)120b)124c)116d)112Correct answer is option 'C'. Can you explain this answer? has been provided alongside types of There are 10 points in a plane, no three of which are in the same straight line, except 4 points which are collinear. The total number of triangles that can be formed with the vertices as these points is:a)120b)124c)116d)112Correct answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice There are 10 points in a plane, no three of which are in the same straight line, except 4 points which are collinear. The total number of triangles that can be formed with the vertices as these points is:a)120b)124c)116d)112Correct answer is option 'C'. Can you explain this answer? tests, examples and also practice CAT tests.
|
|
Explore Courses for CAT exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup