Solution:

Since no two of the $40$ lines are parallel, every two lines cross, giving $\binom{40}{2} = 780$ pairs of lines, and each pair meets at exactly one point. A point where exactly $k$ lines meet accounts for exactly $\binom{k}{2}$ of these pairs.

The given points account for $3\binom{3}{2} + 4\binom{4}{2} + 5\binom{5}{2} + 6\binom{6}{2} = 9 + 24 + 50 + 90 = 173$ pairs of lines. Each remaining pair meets at a point where exactly $2$ lines intersect, one point per pair, so there are $780 - 173 = 607$ such points.