Permutations & Combinations
Intersection Points of Lines — Parallel and Concurrent
nta_pyq_2024_jan
Grade 11
Question:
The lines $L_1,L_2,\ldots,L_{20}$ are distinct. For $n=1,2,3,\ldots,10$ 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 $\{L_1,L_2,\ldots,L_{20}\}$ is equal to:
Step-by-Step Solution
Key Concept: $L_1,L_3,\ldots,L_{19}$ (10 lines) are parallel — 0 intersections among them. $L_2,L_4,\ldots,L_{20}$ (10 lines) are concurrent at $P$ — only 1 intersection point among them instead of $\binom{10}{2}=45$. Total $=\binom{20}{2}-\binom{10}{2}-\binom{10}{2}+1$.
$\binom{20}{2}-\binom{10}{2}-\binom{10}{2}+1=190-45-45+1=101$.
Correct Answer: 101