There are 50 distinct lines $L_1, L_2, ..., L_{50}$. All lines $L_{4n}$, where n a positive integer, are parallel to each other. All lines $L_{4n-3}$, where n is a positive integer, pass through a point P. The maximum possible number of points of intersection of pairs of lines in the set $\{L_1, L_2, ...\}$ is

Looking at the structure of the 50 lines, let me first identify which lines belong to each category.

Lines $L_{4n}$ where $n$ is a positive integer (all parallel to each other):

$L_4, L_8, L_{12}, L_{16}, L_{20}, L_{24}, L_{28}, L_{32}, L_{36}, L_{40}, L_{44}, L_{48}$

This gives us 12 parallel lines (for $n = 1, 2, 3, ..., 12$).

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Lines $L_{4n-3}$ where $n$ is a positive integer (all pass through point P):

$L_1, L_5, L_9, L_{13}, L_{17}, L_{21}, L_{25}, L_{29}, L_{33}, L_{37}, L_{41}, L_{45}, L_{49}$

This gives us 13 lines through point P (for $n = 1, 2, 3, ..., 13$).

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The remaining lines with no special constraints:

$L_2, L_3, L_6, L_7, L_{10}, L_{11}, L_{14}, L_{15}, L_{18}, L_{19}, L_{22}, L_{23}, L_{26}, L_{27}, L_{30}, L_{31}, L_{34}, L_{35}, L_{38}, L_{39}, L_{42}, L_{43}, L_{46}, L_{47}, L_{50}$

This gives us 25 lines (since $50 - 12 - 13 = 25$).

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To maximize the number of intersection points:

The 12 parallel lines don't intersect each other: 0 points

The 13 lines through point P all intersect at P: 1 point (not 13 choose 2, since they all meet at the same point)

The 12 parallel lines intersecting with the 13 lines through P:

$12 \times 13 = 156$ points

The 25 remaining lines should be positioned to intersect with all other lines at distinct points:

Intersections among themselves: $\binom{25}{2} = \dfrac{25 \times 24}{2} = 300$ points

Intersections with the 12 parallel lines: $25 \times 12 = 300$ points

Intersections with the 13 lines through P: $25 \times 13 = 325$ points

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Total number of intersection points:

$1 + 156 + 300 + 300 + 325 = 1082$

Therefore, the maximum possible number of intersection points is 1082.