Question

There are $n$ points in a plane out of these points no three are in the same straight line except $p$ points which are collinear. Let "$k$" be the number of straight lines and "$m$" be the number of triangles .Then find $m-k$ ?(Assume $n=7p=5$)

Answer 1

$\left(i\right)$ Number of lines formed any two points out of given $N$ points $={}^n{C}_2$ and number of lines formed by joining any two points out of $p$ collinear points $={}^p{C}_2$. But these collinear points giving exactly one straight line passing through all of them.
Hence required number of straight lines $={}^n{C}_2-{}^n{C}_2+1$
$\left(ii\right)$ Number of triangles formed by joining any three points out of given $n$ points $={}^n{C}_3$ and number of triangles formed by joining any three points out of $p$ collinear points $={}^p{C}_3$. But no triangle would be formed by joining any three points out of these $p$ collinear points.
Hence the number of triangles formed $={}^n{C}_3-{}^p{C}_3$