Question

Show if each of m points in one straight line be joined to each of n in another by straight lines terminated by the points, then excluding the given points, the lines will intersect $\frac{1}{4}$ mn(m - 1) (n - 1) times.

Answer 1

Let $A_1,A_2,A_3,...,A_n$ and $B_1,B_2,B_3,...,B_m$ be the points on the two given lines. Then it can be seen that
$A_2{B}_1$ will intersect $(m-1)$ lines originating from $A_1$.
$A_3{B}_1$ will intersect $2(m-1)$ lines originating from $A_1$,$A_2$
.........
$A_n{B}_1$ will intersect $(n-1)(m-1)$ lines originating from $A_1,A_2,A_3,...,A_{n-1}$
Next, $A_2{B}_2$ will intersect $(m-2)$ lines originating from $A_1$
$A_3{B}_2$ will intersect $2(m-2)$ lines originating from $A_1$,$A_2$.
$A_n{B}_2$ will intersect $(n-1)(m-2)$ lines originating from $A_1,A_2,A_3,...,A_n$
Similarly, for other points
Now considering all the m points
$B_1,B_2,B_3,...,B_n$ in succession, we get number of points of intersection
$=[1+2+3+...+(n-1)][(m-1)+(m-2)+...+3+2+1]$
$=\frac{n(n-1)}{2}\times \frac{m(m-1)}{2}$
$=\frac{1}{4}mn(m-1)(n-1)$