24. There are $n$ distinct lines in the plane. One of these lines intersects exactly 5 of the $n$ lines, another of these intersects exactly 9 of the $n$ lines, and yet another intersects exactly 11 of them. Which of the following is the smallest possible value of $n$ ?
A 12
B 13
C 14
D 25
E 28
Solution
A

Since one of the lines intersects 11 other lines, the minimum number of lines is at least 12 . To show that exactly twelve lines can satisfy the requirement, consider a family of seven parallel lines, another family of three parallel lines not parallel to the first family and two other lines, intersecting each other and intersecting both families of parallel lines, as shown.

Any line in the family of seven intersects 5 other lines, any of the lines in the second family of three parallel lines intersects 9 other lines, either of two lines not in either family intersects 11 other lines.
The diagram shows a possible construction with this minimum number of lines. Therefore, the answer is 12 .