Please show work and fully answer the question

Question

Five points, not necessarily different, are selected on the number line between and possibly including 0 and 1. After I calculate the distance between each pair of points, I add all ten distances. What is the greatest possible value of this sum?

Answer 1

5 points, each pair with maximum distance apart = 1

sum 10 such distances = 10x1 = 10

if there's only 5 points, then the pairs are found using the handshake theorem

or draw a pentagon and count how many lines connect the vertices

a triangle has 3 vertices and 3 lines connecting them

a square has 4 vertices and 6 lines connecting them

the 4 sides plus 2 diagonals

a pentagon has 5 vertices and 5 diagonals = 10 pairs

BUT with 5 points some pairs on a straight linear line would have zero distance between them, while others would have the max distance of 10

the problem is a little trickier than it seems.

put one point in the middle, total sum of distances = 4 +4(1/2) = 6

at least 6, maybe more, but less than 10

if you places the 5 points equidistant, in a row, with 1/4 between each subsequent pair, the total distance is only 5

Answer 2

The two poinst farthest apart are (0, 0) and (1, 1), Draw it and you shall see.

The distance is √2.

Since all five points can be the same, the total is 5√2