Dhaka Higher Secondary 2010/11
Posted: Tue Jan 18, 2011 2:07 pm
$n$ points are taken on each side of a regular $m$ gon. What is the total number of straight lines that can be drawn using all those points?(except the sides of $m$ gon)
The solution was posted here (it was a regular n gon with m points on each side).
(From the post)
If we choose any two points that are not on same side of the $n\ gon$ then we will find a straight line.
Lets number the sides of $n\ gon$ with $1,2,3,...,n$.
Start with side $1$. For each of the $m$ points of this side, we have $(n-1)m$ points. So in total $(n-1)m^2$ lines.
For each point of side #$2$, we have $(n-2)m$ points. So $(n-2)m^2$ lines.
Thus, total number of line is\[(n-1)m^2+(n-2)m^2+(n-3)m^2+...+1\cdot m^2\] \[=\frac{n(n-1)}{2}\cdot m^2\]