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Out of 15 points in a plane, no three are in a straight line except 8 points which are collinear. How many triangles can be formed by joining them?
- a)504
- b)552
- c)560
- d)1120
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Out of 15 points in a plane, no three are in a straight line except 8 ...
Solution :- No. of Δs = 15C3 - 8C3
= 15!/(3!*12!) - (8!/(3!*5!)
= 455 - 56 = 399
Most Upvoted Answer
Out of 15 points in a plane, no three are in a straight line except 8 ...
Given information
- 15 points in a plane
- 8 points are collinear (lie on a straight line)
- No three points are collinear (do not lie on a straight line)
To find
- Number of triangles that can be formed by joining the points
Approach
- Use the formula for combinations to find the number of triangles that can be formed
- Subtract the number of triangles that have at least one collinear point from the total number of triangles
- Add the number of triangles that have all three collinear points back to the result
Calculations
- Total number of triangles that can be formed using 15 points = 15C3 = 455
- Number of triangles that have at least one collinear point = 8C1 * 7C2 + 7C1 * 6C2 + 6C1 * 5C2 + 5C1 * 4C2 = 1540
- Choose one collinear point and two non-collinear points from the remaining 7 points, and add up for all 8 collinear points
- Number of triangles that have all three collinear points = 8C3 = 56
- Number of triangles that can be formed using non-collinear points = 455 - 1540 + 56 = 399
Answer
- The number of triangles that can be formed by joining the points is 399
- Option (a) is correct.
- 15 points in a plane
- 8 points are collinear (lie on a straight line)
- No three points are collinear (do not lie on a straight line)
To find
- Number of triangles that can be formed by joining the points
Approach
- Use the formula for combinations to find the number of triangles that can be formed
- Subtract the number of triangles that have at least one collinear point from the total number of triangles
- Add the number of triangles that have all three collinear points back to the result
Calculations
- Total number of triangles that can be formed using 15 points = 15C3 = 455
- Number of triangles that have at least one collinear point = 8C1 * 7C2 + 7C1 * 6C2 + 6C1 * 5C2 + 5C1 * 4C2 = 1540
- Choose one collinear point and two non-collinear points from the remaining 7 points, and add up for all 8 collinear points
- Number of triangles that have all three collinear points = 8C3 = 56
- Number of triangles that can be formed using non-collinear points = 455 - 1540 + 56 = 399
Answer
- The number of triangles that can be formed by joining the points is 399
- Option (a) is correct.
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Out of 15 points in a plane, no three are in a straight line except 8 points which are collinear. How many triangles can be formed by joining them?a)504b)552c)560d)1120Correct answer is option 'A'. Can you explain this answer?
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