There are `n` points in a plane in which no large no three are in a straight line except `m` which are all i straight line. Find the number of (i) different straight lines, (ii) different triangles, (iii) different quadrilaterals that can be formed with the given points as vertices.

Explanation

A straight line can be formed by joining any two points. So number of straight lines (i.e., selection of two points from n) is `.^(n)C_(2)` But, selection of two points from m collinear points gives no extra line. Hence, the number of distinct straight lines is
`.^(n)C_(2)- (.^(m)C_(2)-1)=(1)/(2)n(n-1)-(1)/(2)m(m-1)+1`
(b) Formation of traingles is equivalents to section of three points from a points. As m points are collinear, selection of three points from m collinear points from m collinear points gives no triangles. Hence, the number of triangles is
`.^(n)C_(3)- .^(m)C_(3)=(1)/(6)[n(n-1)(n-2)-m(m-1)(m-2)]`
(c ) Four points determine a quadrilateral. But of these four points, not more than two is to be selected from m collinear points. Now the numebr of selections of four points from all n is `.^(n)C_(4)`. The number of selections of three points from m collinear and one from rest is `.^(m)C_(3) .^(n-m)C_(1)`. The number of selections of four points from m collinear is `.^(m)C_(4)`.
Hence, Number of quadrilaterals `= .^(n)C_(4)- .^(m)C_(3)xx (n-m)C_(1)- .^(m)C_(4)`