Question

Given a collection of points p in the plane , a 1-set is a point in p that can be separated from the rest by a line, .i.e the point lies on one side of the line while the others lie on the other side. the number of 1 -sets of p is denoted by n1(p). the minimum value of n1(p) over all configurations p of 5 points in the plane in general position (.i.e no three points in p lie on a line) is

Answer 1

For Maximize: The number of methods, Points should be drawn in the circumference of the CIRCLE So answer would be the number of total point in the plane.

For minimize: The number of methods we should draw 3 points in a TRIANGLE and all other point inside this triangle in such a way that no 3 points could be in a line.

If it will be allowed to draw the points in a line then minimum possibilities will be 2 only, because if we take all points in a line then you can separate only the corner points from the others.

For example

If we have 10 points then maximum possibilities = 10

and minimum = 3

Similarly for 5 points  

Max = 5, Min = 3

For 19  

Max= 19 Min = 3