# Amount of lengths between k points

• B
• Einstein's Cat
In summary, the total number of lengths between k points on a straight line can be calculated by using the formula ##\binom k 2 = \frac{k!}{2! (k-2)!}##, where k is the number of points. This pattern can be observed by looking at examples for k = 2, 3, 4, and 5 points. However, the specific location of the points needs to be specified in order to accurately calculate the distances.
Einstein's Cat
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?

Einstein's Cat said:
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?
What do you get if there are three points? Four? Five?

BTW, it's probably reasonable to assume that the points are all distinct.

Mark44 said:
What do you get if there are three points? Four? Five?

BTW, it's probably reasonable to assume that the points are all distinct.
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

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:
Einstein's Cat
Einstein's Cat said:
however k=4 gave 6 lengths and k=5 gave 10, this defies such a pattern surely?
All is well

Einstein's Cat said:
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
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.

## 1. How is the amount of lengths between k points calculated?

The amount of lengths between k points is calculated by finding the distance between each point and then adding them together. This can be done using various mathematical formulas, such as the Pythagorean theorem or the distance formula.

## 2. What factors can affect the amount of lengths between k points?

The amount of lengths between k points can be affected by the location and positioning of the points, as well as any obstacles or barriers between them. Additionally, the measurement units used and any rounding or approximation can also impact the calculated amount of lengths.

## 3. Can the amount of lengths between k points be negative?

No, the amount of lengths between k points cannot be negative. Length is a measure of distance and is always a positive value. If the calculated amount of lengths is negative, it is likely due to an error in measurement or calculation.

## 4. How does the number of points affect the amount of lengths between them?

The number of points does not necessarily affect the amount of lengths between them. Each point will still have a distance to every other point, but the total amount of lengths may increase as the number of points increases. For example, if there are 3 points, there will be 3 distances calculated. But if there are 5 points, there will be 10 distances calculated.

## 5. Can the amount of lengths between k points be used to determine the shape or pattern of the points?

No, the amount of lengths between k points alone cannot determine the shape or pattern of the points. Other factors, such as the coordinates of the points or the arrangement of the points, are also needed to determine the shape. The amount of lengths between k points can provide information about the distances between points, but it is not enough to determine the shape.

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