Question

In a plane there are 37 straight lines of which 13 pass through the point A and 11 pass through the point B. Besides, no three lines pass through one point, no line passes through both points A and B, and no two are parallel. Find the number of points of intersection of the straight lines.

Answer 1

Hint: First of all consider all the lines to be intersecting at different points and find the number of points of intersection using the combination formula of $^n{C_2}$. Then remove the cases of 13 lines which are passing through point A and another 11 lines which are passing through point B.

Complete step by step solution:
According to the question, there are 37 straight lines of which 13 pass through point A and 11 pass through point B. Other than this, no three lines pass through the same point, no line passes through both points A and B and no two of them are parallel.
Let the total number of intersection of points is denoted by $N$.
First, if we are given 37 non-parallel lines, all of which intersect at different points then the number of points of intersection can be obtained by the formula of $^n{C_2}$. So we get:
$
   \Rightarrow N{ = ^{37}}{C_2} \\
   \Rightarrow N = \dfrac{{37 \times 36}}{2} = 666
 $
But as per the condition given in the question, 13 of those lines are passing through point A. This means that all those 13 lines have only 1 point of intersection. So we will remove the point of intersection of 13 lines and add only 1 for point A. This will give us:
$
   \Rightarrow N = 666{ - ^{13}}{C_2} + 1 \\
   \Rightarrow N = 666 - \dfrac{{13 \times 12}}{2} + 1 \\
   \Rightarrow N = 666 - 78 + 1 = 589
 $
Also 11 other lines (it is given that no line is passing through both A and B) are passing through point B. So we will also do the same calculation for these lines as we did for above case. Now after removing these cases also, the number of points of intersection will be:
$
   \Rightarrow N = 589{ - ^{11}}{C_2} + 1 \\
   \Rightarrow N = 589 - \dfrac{{11 \times 10}}{2} + 1 \\
   \Rightarrow N = 589 - 55 + 1 = 535
 $
Since apart from this, no three lines are passing through the same point and no two lines are parallel, so this is our final answer.

Thus the total number of points of intersection of the lines following the above conditions is 535.

Note: We use the formula for the points of intersection of non-parallel lines which is of combination as we have used above. This formula is also shown below:
${ \Rightarrow ^n}{C_2} = \dfrac{{n\left( {n - 1} \right)}}{2}$, where $n$ is the number of non-parallel lines such that no three of them pass through the same point.
For the arrangement of different objects, we use the formula of permutation. This is shown below:
${ \Rightarrow ^n}{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}$