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There are seven points on a piece of paper. Exactly four of these points are on a straight line. No other line contains more than two of these points. Three of these seven points are selected to form the vertices of a triangle. How many triangles are possible?
Correct answer is '31'. Can you explain this answer?
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There are seven points on a piece of paper. Exactly four of these poi...
There are three cases to consider corresponding to zero vertices, one vertex or two vertices on the given line
Case 1 'zero vertices on the given line'Since there are exactly three points not on the line, there can only be one triangle formed with these three points.
Case 2 'one vertex on the given line'There are four choices for the point on the line and for each of these four points there are three ways of selecting the pair of vertices not on the line. Thus, there are 3 × 4 or 12 possible triangles
Case 3 '2 vertices on the given line'There are six ways of choosing the pair of points on the line and for each of these six pairs there are three ways of selecting the vertex not on the line giving a total of 6 × 3 or 18 possibilities.
In total there are 1+12+18 or 31 triangles
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There are seven points on a piece of paper. Exactly four of these poi...
Breaking Down the Problem:
To solve this problem, we need to consider the different possibilities for forming triangles using three of the seven given points on the paper. Let's break down the problem into smaller parts:
1. Number of Ways to Select 3 Points:
The first step is to determine the number of ways we can select 3 points from the given 7 points. This can be calculated using the combination formula, denoted as C(n, r), where n is the total number of points and r is the number of points to be selected. In this case, we need to calculate C(7, 3).
Using the combination formula:
C(7, 3) = 7! / (3!(7-3)!) = 7! / (3!4!) = (7 * 6 * 5) / (3 * 2 * 1) = 35
So, there are 35 different ways to select 3 points from the given 7 points.
2. Number of Ways to Form a Triangle:
Now, we need to determine how many of these 35 selections of 3 points can actually form a triangle. In order to form a triangle, the selected points must not be collinear (lying on the same straight line) and no other line should contain more than 2 points.
2.1. Number of Collinear Point Combinations:
To count the number of collinear point combinations, we need to consider the number of ways to select 3 points that lie on the same straight line. Since exactly 4 points are on a straight line, we can choose any 3 of those points to form a collinear combination. This can be calculated using the combination formula C(4, 3).
Using the combination formula:
C(4, 3) = 4! / (3!(4-3)!) = 4! / (3!1!) = (4 * 3 * 2) / (3 * 2 * 1) = 4
So, there are 4 collinear point combinations.
2.2. Number of Ways to Form a Triangle:
To find the number of ways to form a triangle, we subtract the number of collinear point combinations from the total number of selections:
Number of ways to form a triangle = Total selections - Collinear point combinations
= 35 - 4
= 31
Therefore, there are 31 possible triangles that can be formed using three of the given seven points on the paper.
To solve this problem, we need to consider the different possibilities for forming triangles using three of the seven given points on the paper. Let's break down the problem into smaller parts:
1. Number of Ways to Select 3 Points:
The first step is to determine the number of ways we can select 3 points from the given 7 points. This can be calculated using the combination formula, denoted as C(n, r), where n is the total number of points and r is the number of points to be selected. In this case, we need to calculate C(7, 3).
Using the combination formula:
C(7, 3) = 7! / (3!(7-3)!) = 7! / (3!4!) = (7 * 6 * 5) / (3 * 2 * 1) = 35
So, there are 35 different ways to select 3 points from the given 7 points.
2. Number of Ways to Form a Triangle:
Now, we need to determine how many of these 35 selections of 3 points can actually form a triangle. In order to form a triangle, the selected points must not be collinear (lying on the same straight line) and no other line should contain more than 2 points.
2.1. Number of Collinear Point Combinations:
To count the number of collinear point combinations, we need to consider the number of ways to select 3 points that lie on the same straight line. Since exactly 4 points are on a straight line, we can choose any 3 of those points to form a collinear combination. This can be calculated using the combination formula C(4, 3).
Using the combination formula:
C(4, 3) = 4! / (3!(4-3)!) = 4! / (3!1!) = (4 * 3 * 2) / (3 * 2 * 1) = 4
So, there are 4 collinear point combinations.
2.2. Number of Ways to Form a Triangle:
To find the number of ways to form a triangle, we subtract the number of collinear point combinations from the total number of selections:
Number of ways to form a triangle = Total selections - Collinear point combinations
= 35 - 4
= 31
Therefore, there are 31 possible triangles that can be formed using three of the given seven points on the paper.
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There are seven points on a piece of paper. Exactly four of these points are on a straight line. No other line contains more than two of these points. Three of these seven points are selected to form the vertices of a triangle. How many triangles are possible?Correct answer is '31'. Can you explain this answer?
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There are seven points on a piece of paper. Exactly four of these points are on a straight line. No other line contains more than two of these points. Three of these seven points are selected to form the vertices of a triangle. How many triangles are possible?Correct answer is '31'. 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 seven points on a piece of paper. Exactly four of these points are on a straight line. No other line contains more than two of these points. Three of these seven points are selected to form the vertices of a triangle. How many triangles are possible?Correct answer is '31'. 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 seven points on a piece of paper. Exactly four of these points are on a straight line. No other line contains more than two of these points. Three of these seven points are selected to form the vertices of a triangle. How many triangles are possible?Correct answer is '31'. Can you explain this answer?.
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