Question

There are 2n points in a plane in which m are collinear (n > m > 4). Number of quadrilateral formed by joining these lines is

• A. Equal to $2nC_{4} -^{m}C_{4}$
• B. Greater than $2nC_{4} -^{m}C_{4}$
• C. Less than $2nC_{4} -^{m}C_{4}$
• D. None of these

Answer 1

Answer: (C)

Less than $2nC_{4} -^{m}C_{4}$

Answer 2

$2nC_{4}$ is the number of ways in which 4 points can be selected out of 2n points.

$mC_{4}$ is the number of ways in which 4 points can be selected out of m points (and in this case quadrilateral will not be formed as m points are collinear).

Also, if we select 3 points from ‘m’ collinear points and 1 point from 2n non collinear points quadrilateral will not be formed such type of selection will be $mC_{3}x^{2n}C_{1}$.

∴ Total quadrilateral = $2nC_{4} -^{m}C_{4} -^{m}C_{3}x^{2n}C_{1}$

Thus, number of quadrilateral is less than $2nC_{4} -^{m}C_{4}$