Question:
Out of 18 points in a plane, no three are in the same line except five points which are collinear. Find the number of lines that can be formed joining the point.
[Hint: Number of straight lines =18C2 – 5C2 + 1]
Solution:
We know that,
nCr
$=\frac{\mathrm{n} !}{\Gamma !(\mathrm{n}-\mathrm{r}) !}$
According to the question,
Number of points = 18
Number of Collinear points = 5
Number of lines form by 18 points = 18C2
For 5 points to be collinear = 5C2
The number of lines that can be formed joining the point, = 18C2–5C2+1
$=\frac{18 !}{2 ! 6 !}-\frac{5 !}{2 ! 3 !}+1$
=153-10+1
=144