Q$2.(15$ points$)$ Consider the following problem:

Five points $($A$,$B$,$C$,$D$,$E$),$ not necessarily different, are selected on a number line so that the distance between any two of these points is at most $1.$ After calculating the distance between each pair of points, I add all ten distances. What is the greatest possible value of this sum?

Your task is not to solve this problem algebraically $($which is tough!$).$ Instead, write a code that takes advantage of the speed of modern computers by checking many possible solutions. For example, you might make a script to build an Nx$6$ matrix with the following constraints:

$$ First column is $0($i$.$e$.$ A $=0)$

$$ Fifth column is $1($i$.$e$.$ E $=1,$ so the distance between A and E is $1)$

$$ Second, third and fourth columns are random real numbers between $0$ and $1$

$$ Sixth column is the sum of the $10$ distances between points $($AB distance $+$ AC distance $+$ AD distance $+$ AE distance $\{$which is $1\}+$ BC distance $+...+$ DE distance$)$

Your output should look like:

Max value is x$.$xxx $.$

The $5$ numbers are: $0,0.$xxx$,0.$xxx$,0.$xxx$,1$

Initially you may want N to be a reasonably small integer so you can look at the data and debug. Once your code is working properly, run your code several times while increasing N as you go $$ look for a pattern. Finally, report the specific output of your code when N is sufficiently large to be sure of your answer.

From the data you generated, replace the X$$s in the sentence below by theorizing a general solution to the original question, and specify $5$ numbers which would give your theorized max sum.

The maximum possible value is X$.$XXX $.$

Five numbers that give this are: $0,0.$xxx$,0.$xxx$,0.$xxx$,1.$

IN MATLAB CODE