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: Start with total intersections.
If all lines were in general position, intersections would be \(\binom{37}{2}\).

Step 2: Adjust for concurrencies at A and B.
The 13 lines through \(A\) give only 1 intersection point instead of \(\binom{13}{2}\), so subtract \(\binom{13}{2}-1\). Similarly for 11 lines through \(B\): subtract \(\binom{11}{2}-1\).

Step 3: Final count.
\[\binom{37}{2} - \binom{13}{2} + 1 - \binom{11}{2} + 1 = \binom{37}{2} - \binom{13}{2} - \binom{11}{2} + 2.\]
\[\boxed{\binom{37}{2} - \binom{13}{2} - \binom{11}{2} + 2}\]