$35 \cdot 41 x$ On the positive half-axis of $x$, there are 10 points, and on the positive half-axis of $y$, there are 5 points. The 10 points on the positive half-axis of $x$ and the 5 points on the positive half-axis of $y$ form 50 line segments. The maximum number of intersection points of these 50 line segments in the first quadrant is
(A) 250 .
(B) 450 .
(C) 500 .
(D) 1250 .
(E) 2500 .
[Sol] Note the following fact:
Points on the positive half of the $x$-axis and points on the positive half of the $y$-axis can form a convex quadrilateral, and their diagonals have one intersection point.
Thus, the lines connecting these points can have at most $C_{10}^{2} \cdot C_{5}^{2}=450$ intersection points.
Therefore, the answer is $(B)$.