Question
Given four 6 collinear points, make a conjecture about the number of ways to connect different pairs of points
The correct answer is: 15 ways to connect different pairs of points
We have given the 6 collinear points
We have to find the number of ways to connect the different pairs of points
We will consider figure of the line
![](data:image/png;base64,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)
We will consider the points from A to F
Here we will consider point A
Point A have 5 options to pair, that are B, C, D, E, F.
If we consider point B then,
Point B have 4 points to pair that are C, D , E, F . Because we have considered the pair with A before.
If we consider point C then,
Point C have 3 points to pair , that are D, E, F .
If we consider point D then
Point D have 2 points to pair , that are E, F.
If we consider point E then
Point E have 1 point to pair, that is F
Therefore total number of ways the points can be paired is = 5 + 4 + 3 + 2 + 1= 15
There are 15 ways to connect different pairs of points.
We will consider the points from A to F
Here we will consider point A
Point A have 5 options to pair, that are B, C, D, E, F.
If we consider point B then,
Point B have 4 points to pair that are C, D , E, F . Because we have considered the pair with A before.
If we consider point C then,
Point C have 3 points to pair , that are D, E, F .
If we consider point D then
Point D have 2 points to pair , that are E, F.
If we consider point E then
Point E have 1 point to pair, that is F
Therefore total number of ways the points can be paired is = 5 + 4 + 3 + 2 + 1= 15
There are 15 ways to connect different pairs of points.