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In a plane, there are 37 straight lines, out of which 13 pass through the point A and 11 pass through the point B. Besides, no three lines pass through the same point, no line passes through both points A and B, and no two lines are parallel. The number of intersection points the lines have is equal to
- a)535
- b)601
- c)728
- d)963
Correct answer is option 'A'. Can you explain this answer?
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In a plane, there are 37 straight lines, out of which 13 pass through ...
Given information:
- There are 37 straight lines in the plane.
- 13 lines pass through point A.
- 11 lines pass through point B.
- No three lines pass through the same point.
- No line passes through both points A and B.
- No two lines are parallel.
Analysis:
To find the number of intersection points, we need to determine how many pairs of lines intersect. We can consider each pair of lines and count the number of intersection points.
Counting the number of pairs of lines:
The number of pairs that can be formed from a set of n lines is given by the formula:
nC2 = n! / (2!(n-2)!)
In this case, we have 37 lines, so the number of pairs of lines is:
37C2 = 37! / (2!(37-2)!) = 37! / (2!35!) = (37 × 36) / 2 = 666
Counting the number of intersection points:
We know that no three lines pass through the same point and no two lines are parallel. This means that each pair of lines will intersect at exactly one point.
So, the number of intersection points is equal to the number of pairs of lines, which is 666.
However, we need to subtract the cases where both lines of a pair pass through either point A or point B, as these points should not be counted as intersection points.
Subtracting the cases with lines passing through point A:
We have 13 lines passing through point A. To find the number of pairs of lines passing through point A, we can use the formula:
13C2 = 13! / (2!(13-2)!) = 13! / (2!11!) = (13 × 12) / 2 = 78
This means that there are 78 pairs of lines passing through point A, which need to be subtracted from the total number of pairs.
Subtracting the cases with lines passing through point B:
Similarly, we have 11 lines passing through point B. To find the number of pairs of lines passing through point B, we can use the formula:
11C2 = 11! / (2!(11-2)!) = 11! / (2!9!) = (11 × 10) / 2 = 55
This means that there are 55 pairs of lines passing through point B, which need to be subtracted from the total number of pairs.
Calculating the final count:
The final count of intersection points is obtained by subtracting the cases with lines passing through point A and point B from the total number of pairs of lines:
666 - 78 - 55 = 533
However, we also need to consider the points A and B as intersection points, as they are the points where multiple lines intersect. So, we need to add 2 to the count.
533 + 2 = 535
Thus, the correct answer is option 'A', 535 intersection points.
- There are 37 straight lines in the plane.
- 13 lines pass through point A.
- 11 lines pass through point B.
- No three lines pass through the same point.
- No line passes through both points A and B.
- No two lines are parallel.
Analysis:
To find the number of intersection points, we need to determine how many pairs of lines intersect. We can consider each pair of lines and count the number of intersection points.
Counting the number of pairs of lines:
The number of pairs that can be formed from a set of n lines is given by the formula:
nC2 = n! / (2!(n-2)!)
In this case, we have 37 lines, so the number of pairs of lines is:
37C2 = 37! / (2!(37-2)!) = 37! / (2!35!) = (37 × 36) / 2 = 666
Counting the number of intersection points:
We know that no three lines pass through the same point and no two lines are parallel. This means that each pair of lines will intersect at exactly one point.
So, the number of intersection points is equal to the number of pairs of lines, which is 666.
However, we need to subtract the cases where both lines of a pair pass through either point A or point B, as these points should not be counted as intersection points.
Subtracting the cases with lines passing through point A:
We have 13 lines passing through point A. To find the number of pairs of lines passing through point A, we can use the formula:
13C2 = 13! / (2!(13-2)!) = 13! / (2!11!) = (13 × 12) / 2 = 78
This means that there are 78 pairs of lines passing through point A, which need to be subtracted from the total number of pairs.
Subtracting the cases with lines passing through point B:
Similarly, we have 11 lines passing through point B. To find the number of pairs of lines passing through point B, we can use the formula:
11C2 = 11! / (2!(11-2)!) = 11! / (2!9!) = (11 × 10) / 2 = 55
This means that there are 55 pairs of lines passing through point B, which need to be subtracted from the total number of pairs.
Calculating the final count:
The final count of intersection points is obtained by subtracting the cases with lines passing through point A and point B from the total number of pairs of lines:
666 - 78 - 55 = 533
However, we also need to consider the points A and B as intersection points, as they are the points where multiple lines intersect. So, we need to add 2 to the count.
533 + 2 = 535
Thus, the correct answer is option 'A', 535 intersection points.
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In a plane, there are 37 straight lines, out of which 13 pass through the point A and 11 pass through the point B. Besides, no three lines pass through the same point, no line passes through both points A and B, and no two lines are parallel. The number of intersection points the lines have is equal toa)535b)601c)728d)963Correct answer is option 'A'. Can you explain this answer?
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