Question

In a plane there are \(37\) straight lines of which \(13\) pass through point \(A\) and \(11\) pass through point \(B\). Moreover, no three lines apart from the lines passing through \(A\) and \(B\) pass through the same point and no two are parallel. What is the number of points of intersection of the straight lines?

• A. \({}^{37}C_{2}\)
• B. \({}^{37}C_{2}-{}^{13}C_{2}-{}^{11}C_{2}\)
• C. \({}^{37}C_{2}-{}^{13}C_{2}-{}^{11}C_{2}+2\)
• D. \({}^{37}C_{2}-2\)

Hint:

For \(n\) lines with no two parallel and no three concurrent, the number of intersection points is \({}^{n}C_{2}\). If \(r\) lines are concurrent, replace \({}^{r}C_{2}\) counted intersections by only \(1\) actual intersection point.

Answer 1

The Correct Option is C

Step 1: Find total possible intersections.
If no two lines are parallel and no three lines are concurrent, then every pair of lines gives one point of intersection.
So, for \(37\) straight lines, the total number of intersections would be \[ {}^{37}C_{2} \]

Step 2: Adjust for lines passing through point \(A\).
There are \(13\) lines passing through point \(A\).
If these \(13\) lines were not concurrent, they would produce \[ {}^{13}C_{2} \] intersection points.
But actually, all these \(13\) lines meet at the same point \(A\).
So instead of \({}^{13}C_{2}\) points, they give only \(1\) point.
Hence, the excess counted intersections for point \(A\) are corrected by subtracting \[ {}^{13}C_{2} \] and adding back \(1\).

Step 3: Adjust for lines passing through point \(B\).
Similarly, \(11\) lines pass through point \(B\).
If these \(11\) lines were not concurrent, they would produce \[ {}^{11}C_{2} \] intersection points.
But actually, all these \(11\) lines meet at the same point \(B\).
So we subtract \[ {}^{11}C_{2} \] and add back \(1\).

Step 4: Write the final expression.
Therefore, the required number of points of intersection is \[ {}^{37}C_{2}-{}^{13}C_{2}+1-{}^{11}C_{2}+1 \] \[ ={}^{37}C_{2}-{}^{13}C_{2}-{}^{11}C_{2}+2 \]

Step 5: Final conclusion.
Hence, the number of points of intersection is \[ \boxed{{}^{37}C_{2}-{}^{13}C_{2}-{}^{11}C_{2}+2} \]