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In a plane, there are 35 straight lines, 17 of which pass through point A and 15 through point B. No line also passes through both points A and B, no two lines are parallel, and no three lines cross at the same location. Find the number of intersections the straight lines have.

Option: 1

254


Option: 2

413


Option: 3

356


Option: 4

525


Answers (1)

best_answer

Given that,

In a plane, there are 35 straight lines, 17 of which and 15 of which pass directly through points A and B.

No two lines are parallel, no three lines intersect a single point, and no line crosses both points A and B.

Let N stand for the total number of intersections between points.

First, the formula for \mathrm{^{n}C_{2}} can be used to determine the number of places of intersection if 35 non-parallel lines are supplied, each of which intersects at a different location.

Thus,

\mathrm{\begin{aligned} & N={ }^{35} C_2 \\ & N=\frac{35 !}{2 ! 33 !} \\ & N=\frac{35 \times 34}{2} \\ & N=595 \end{aligned}}

From the given, 15 of those lines pass through point A in accordance with the conditions stated in the question. This indicates that there is only one point at which all 15 lines intersect. Therefore, we will eliminate the intersection of 15 lines and add just 1 for point A. This is given by,

\mathrm{\begin{aligned} & N=595-{ }^{17} C_2+1 \\ & N=595-\frac{17 !}{2 ! 15 !}+1 \\ & N=595-\frac{17 \times 16}{2}+1 \\ & N=595-136+1 \\ & N=460 \end{aligned}}

Additionally, 15 additional lines pass through point B (given that no line passes through both A and B). Therefore, we will calculate these lines in the same way that we did for the case above. The number of points of intersection will be as follows once these situations have also been eliminated:

\mathrm{\begin{aligned} & N=460-{ }^{15} C_2+1 \\ & N=460-\frac{15 !}{2 ! 13 !}+1 \\ & N=460-\frac{15 \times 14}{2}+1 \\ & N=356 \end{aligned}}

Therefore, the number of lines the straight lines intersect is 356.

 

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