The lines $\mathrm{L}_1, \mathrm{~L}_2, \ldots, \mathrm{I}_{20}$ are distinct. For $\mathrm{n}=1,2,3, \ldots, 10$ all the lines $\mathrm{L}_{2 \mathrm{n}-1}$ are parallel to each other and all the lines $\mathrm{L}_{2 \mathrm{n}}$ pass through a given point $P$. The maximum number of points of intersection of pairs of lines from the set $\left\{L_1, L_2, \ldots, L_{20}\right\}$ is equal to :
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Solution
$\mathrm{L}_1, \mathrm{~L}_3, \mathrm{~L}_5,--\mathrm{L}_{19}$ are Parallel
$\mathrm{L}_2, \mathrm{~L}_4, \mathrm{~L}_6,-\mathrm{L}_{20}$ are Concurrent
Total points of intersection $={ }^{20} \mathrm{C}_2-{ }^{10} \mathrm{C}_2-{ }^{10} \mathrm{C}_2+1 =101$
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