1.64 Let there be $m$ and $n$ fixed points on the positive half-axes of $X$ and $Y$, respectively. Connecting each pair of these $m$ and $n$ points with a line segment, if no three line segments intersect at the same point, determine the number of intersection points of all the line segments (excluding the endpoints of the line segments).

 untranslated text remains unchanged:
设 $X, Y$ 的正半轴上各有 $m$ 个和 $n$ 个定点,在此 $m$ 及 $n$ 个点中, 每两点连一线段,如果每三条线段不共点, 试确定所连全体线段交点的个数(不包括线段的端点). 

(However, the above is the translation of the problem statement into English, while the untranslated text is provided as requested.)
[Solution] Take two points each on the $X$-axis and $Y$-axis to form a convex quadrilateral, the intersection of its diagonals must be in the $\mathrm{I}$ quadrant. Conversely, each intersection point of any line segments corresponds to a convex quadrilateral with two sides on the $X$-axis and $Y$-axis, respectively.

Given that no three line segments intersect at the same point in the $\mathrm{I}$ quadrant, the intersection points of the connected line segments are uniquely determined.
The number of convex quadrilaterals formed by any two points on the $X$-axis and any two points on the $Y$-axis is
$$
C_{m}^{2} C_{n}^{2}=\frac{m n(m-1)(n-1)}{4}
$$

which is the number of intersection points as stated in the problem.