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Combinatorics Difficulty 5.0 AIME, harder Find the answer United Kingdom

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) $28$

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