24 lines are drawn in a plane. Of these, 8 lines pass through point \(A\) and 11 lines pass through point \(B.\) No othe

Vincen wrote: ↑

Tue Jun 30, 2020 6:51 am

24 lines are drawn in a plane. Of these, 8 lines pass through point \(A\) and 11 lines pass through point \(B.\) No other three lines pass through the same point. Also, no line passes through both points \(A\) and \(B\), and no two lines are parallel. Including \(A\) and \(B,\) what is the total number of points of intersections among the 24 lines?

A. 105
B. 193
C. 195
D. 257
E. 276

[spoiler]OA=C[/spoiler]

Solution:

Each of the 8 lines passing through A will intersect each of the 11 lines passing through B at 8 x 11 = 88 points.

There are 8 + 11 = 19 lines that pass through A or B. The 20th line will intersect these 19 lines in 19 points, the 21st line will intersect the previous 20 lines in 20 points, and so on. Thus, the 20th, 21st, 22nd, 23rd and 24th lines will create 19 + 20 + 21 + 22 + 23 = 105 intersection points.

Adding the number of intersections (and adding 2 for the points A and B), we see that there are 88 + 105 + 2 = 195 intersection points.

Answer: C