Number of lines from 11 points of which 5 are collinear.

Question

I came across a question that goes as follows:

Find the number of straight lines if there are 11 points in a plane of which 5 are collinear.

The following is how I approached the problem:

Since 5 of the 11 points are collinear, 6 points are non-collinear. We can use these 6 points AND one point from those 5 collinear points to make our lines, a total of 7 points. And from these 7 points, we can choose 2 to make our straight lines in 7C2 ways.

This however is not the right answer. The right answer is 46.

Could someone explain where I am going wrong? Approaches better than this are also welcome!

Thanks!

Answer 1

Let the points be $A_1,A_2,\cdots,A_{10},A_{11}$ and suppose $A_1,A_2,A_3,A_4,A_5$ be collinear. In your approach you replaced these $5$ points with another point say $B$. However this approach does not cover all the lines because $A_1A_{10}$, $A_2A_{10}$ produce different lines but if you replace $A_1,A_2,A_3,A_4,A_5$ with $B$ you get only one line.

We can solve this as follows.
We can choose $2$ points in $^{11}C_2$ ways. Out of these pairs of point $^{5}C_2$ pairs of point give the same line. So the total number of lines that can be formed is $^{11}C_2-^{5}C_2+1=55-10+1=46$.

Answer 2

First there's 1 line which goes through 5 collinear points. Second, there's 6x5=30 lines which connect each point of the 6 non-collinear points and each point of the 5 collinear points. Finally, there's $C^2_7=15$ lines among the 6 non-collinear points. So there's a total of 46 lines.