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B Amount of lengths between k points

  1. Dec 8, 2016 #1
    Say that there lies k points upon a straight line.

    My question is this: what is the total amount of lengths between these points?

    To elaborate, say k is 2; there are two points and so there is only one length between the two points. How many lengths would there be with k > 2?

    What is the proof for a solution to this?
     
  2. jcsd
  3. Dec 8, 2016 #2

    Mark44

    Staff: Mentor

    What do you get if there are three points? Four? Five?

    BTW, it's probably reasonable to assume that the points are all distinct.
     
  4. Dec 8, 2016 #3
    I can represent each length between points like this for k= 3 (each pair of integers represents the length between the points that are labelled such)

    1, 2
    1,3
    2,3

    for k= 4:

    1,2
    1,3
    1,4
    2,3
    2,4
    3,4

    for k= 5

    1,2
    1,3
    1,4
    1,5
    2,3
    2,4
    2,5
    3,4
    3,5
    4,5
     
  5. Dec 8, 2016 #4

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    Going from k=2 with 1 distance to k=3, you got 2 additional distances (those to the additional point). Going to k=4, you got 3 additional distances. Do you see a pattern?

    Edit: Typo
     
    Last edited: Dec 8, 2016
  6. Dec 8, 2016 #5
    All is well
     
  7. Dec 8, 2016 #6

    Mark44

    Staff: Mentor

    You need to be more specific about where the points are located. In your example for k = 3, the points appear to be at 1, 2, and 3 units, respectively. For three points, there are ##\binom 3 2## ways (the number of combinations of 3 things taken 2 at a time) of choosing two points among the three, where ##\binom 3 2 = \frac{3!}{2! 1!}## . For four points, there are ##\binom 4 2 = \frac{4!}{2! 2!}## = 6 ways. For each pairing of points, you need to calculate the distance between the two points.
     
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