The lines L_1, L_2, , L_20 are distinct.

Question

The lines $L_1, L_2, \ldots, L_{20}$ are distinct. For $n=1,2,\ldots,10$, all the lines $L_{2n-1}$ are parallel to each other, and all the lines $L_{2n}$ pass through a given point $P$. The maximum number of points of intersection of pairs of lines from the set $\{L_1,\ldots,L_{20}\}$ is equal to:

Answer: 101

Accepted Answer

There are 10 mutually parallel lines ($L_1,L_3,\ldots,L_{19}$) contributing no intersections among themselves, and 10 concurrent lines ($L_2,L_4,\ldots,L_{20}$) all meeting at $P$, contributing just $1$ point among themselves.

Intersections among the 10 parallel lines: $0$.

Intersections among the 10 concurrent lines: $1$ (all meet at $P$).

Intersections between a parallel line and a concurrent line: each of the $10$ parallel lines meets each of the $10$ concurrent lines at a distinct point (generically): $10\times10=100$.

$$\text{Total} = 0+1+100 = 101$$

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