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How many straight lines can be drawn with 15 points on a plane of which 7 are collinear?
- a)105
- b)84
- c)85
- d)90
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
How many straight lines can be drawn with 15 points on a plane of whic...
To find the number of straight lines that can be drawn with 15 points on a plane, we need to consider two cases: points that are collinear and points that are not collinear.
Case 1: Points that are collinear
Given that there are 7 collinear points, we can form straight lines by choosing any 2, 3, 4, 5, or 6 points from the collinear set.
- Choosing 2 points from the collinear set: This can be done in 7C2 ways, which is equal to 7! / (2! * (7-2)!) = 21 ways.
- Choosing 3 points from the collinear set: This can be done in 7C3 ways, which is equal to 7! / (3! * (7-3)!) = 35 ways.
- Choosing 4 points from the collinear set: This can be done in 7C4 ways, which is equal to 7! / (4! * (7-4)!) = 35 ways.
- Choosing 5 points from the collinear set: This can be done in 7C5 ways, which is equal to 7! / (5! * (7-5)!) = 21 ways.
- Choosing 6 points from the collinear set: This can be done in 7C6 ways, which is equal to 7! / (6! * (7-6)!) = 7 ways.
Therefore, the total number of straight lines that can be formed by choosing points from the collinear set is 21 + 35 + 35 + 21 + 7 = 119.
Case 2: Points that are not collinear
For the remaining 8 points that are not collinear, any 2 points can form a straight line.
Choosing 2 points from the non-collinear set: This can be done in 8C2 ways, which is equal to 8! / (2! * (8-2)!) = 28 ways.
Therefore, the total number of straight lines that can be formed by choosing points from the non-collinear set is 28.
Total number of straight lines = Number of straight lines from collinear set + Number of straight lines from non-collinear set
Total number of straight lines = 119 + 28 = 147.
However, this calculation includes some duplicate lines where both points are from the collinear set. Since we want to find unique straight lines, we need to subtract the number of duplicate lines.
Number of duplicate lines = Number of lines formed by choosing 2 points from the collinear set
Number of duplicate lines = 7C2 = 7! / (2! * (7-2)!) = 21
Therefore, the total number of unique straight lines that can be formed with 15 points on a plane, of which 7 are collinear, is 147 - 21 = 126.
So, the correct answer is option 'C' (85).
Case 1: Points that are collinear
Given that there are 7 collinear points, we can form straight lines by choosing any 2, 3, 4, 5, or 6 points from the collinear set.
- Choosing 2 points from the collinear set: This can be done in 7C2 ways, which is equal to 7! / (2! * (7-2)!) = 21 ways.
- Choosing 3 points from the collinear set: This can be done in 7C3 ways, which is equal to 7! / (3! * (7-3)!) = 35 ways.
- Choosing 4 points from the collinear set: This can be done in 7C4 ways, which is equal to 7! / (4! * (7-4)!) = 35 ways.
- Choosing 5 points from the collinear set: This can be done in 7C5 ways, which is equal to 7! / (5! * (7-5)!) = 21 ways.
- Choosing 6 points from the collinear set: This can be done in 7C6 ways, which is equal to 7! / (6! * (7-6)!) = 7 ways.
Therefore, the total number of straight lines that can be formed by choosing points from the collinear set is 21 + 35 + 35 + 21 + 7 = 119.
Case 2: Points that are not collinear
For the remaining 8 points that are not collinear, any 2 points can form a straight line.
Choosing 2 points from the non-collinear set: This can be done in 8C2 ways, which is equal to 8! / (2! * (8-2)!) = 28 ways.
Therefore, the total number of straight lines that can be formed by choosing points from the non-collinear set is 28.
Total number of straight lines = Number of straight lines from collinear set + Number of straight lines from non-collinear set
Total number of straight lines = 119 + 28 = 147.
However, this calculation includes some duplicate lines where both points are from the collinear set. Since we want to find unique straight lines, we need to subtract the number of duplicate lines.
Number of duplicate lines = Number of lines formed by choosing 2 points from the collinear set
Number of duplicate lines = 7C2 = 7! / (2! * (7-2)!) = 21
Therefore, the total number of unique straight lines that can be formed with 15 points on a plane, of which 7 are collinear, is 147 - 21 = 126.
So, the correct answer is option 'C' (85).
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Community Answer
How many straight lines can be drawn with 15 points on a plane of whic...
If we select any 2 points, we can draw a straight line.
So, total number of straight lines = 15C2
But, we cannot draw 7C2 lines using the 7 collinear points, we can draw only one.
So, we'll subtract those cases from the total number of cases.
Finally, number of straight line that can be drawn = 15C2 - 7C2 + 1 = 85
So, total number of straight lines = 15C2
But, we cannot draw 7C2 lines using the 7 collinear points, we can draw only one.
So, we'll subtract those cases from the total number of cases.
Finally, number of straight line that can be drawn = 15C2 - 7C2 + 1 = 85
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