Four points lie in the same plane, and each point can act as an endpoint. How many unique line segments can be formed using the set of points

Question

This is about points on lines and I have no idea the avaialabe answers are

 

 

A. 8

B. 6

C. 5

D. 4

Answer 1

the assumption is at most 2 points lie on a unique line. In other words, we can not draw a straight line through 3 points or 4 points. So draw points A, B, C, and D at the corners of a square for example.

Starting with A draw a line segment to each of the other points. There are 3.

 

Now starting with B do the same and we get 2 line segments because we can't draw one to point A. This is because segment AB is the same as segment BA.

 

Finally draw a segment between C and D.

 

There are a total of 6 line segments.

 

Using the principle of combinations if you have 4 elements and you combine them two at a time and order does not matter, then the number of combinations possible is 4! / 2! 2! = 4*3*2*1 / 2*1*2*1 = 24 / 4 = 6

Answer 2

The number of segments is equal to the number of couples produced by the set (each segment is defined by two points). The same pair of points produces two identical segments, the number of each is 4x3 = 12. To get unique lines you have to divide this number by 2. Thus, the number of the unique lines is 6. This is a number of combinations of 2 elements  in a set of 4 elements: C²₄= 4x3/2. The answer is B.