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There are $10$ points in a plane of which $4$ are collinear. Then the number of straight lines formed by joining these points is equal to ?

I understand that number of ways of forming a straight line will be equal to $\binom{10}{2}$. However it is said that $4$ points can form only one single line . How is that possible ? I understand that since the points are collinear , I cannot join the first collinear point to the third or fourth collinear point to get a line , but I could stil join the first collinear point to the second , and the third collinear point to the fourth without joining the second and third collinear points to get two lines right ?

Aditi
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    "I cannot join the first colinear point to the third or fourth to get a line" you seem to be thinking of line segments, not lines. If you join the first and the third you get a line which passes through all four. If you join the first and the fourth you get the very same line which passes through all four. If you want some way to visualize the points, take a semicircle and put those four points all on the straight section and the remaining six points around the circular section. – JMoravitz Oct 31 '17 at 14:31
  • Ohh okay ! That rings a bell . Thank you ! – Aditi Oct 31 '17 at 14:35

2 Answers2

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Any pair of four collinear points defines the same line. But $\binom {10}2$ counts all these lines as distinct - it counts lines by counting pairs of points, and this creates multiple counting when different pairs of points define the same line. So you have to adjust for the multiple counting of the same line - how many times is the line with four points counted?

Mark Bennet
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Remember, a line extends infinitely in both directions, else it would be called a segment. So if four points are collinear then they are contained in the same line.

The total number of lines joining two of the points is $\binom{10}{2}$. However, since four of the points are collinear, we have to reduce all the combinations generated by them to a single one. So then the total number of straight lines will be given by

$$\binom{10}{2} - \binom{4}{2} + 1$$

Bergson
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