# Out of 18 points in a plane, no three are in the same straight line except five points which are collinear.

Question:

Out of 18 points in a plane, no three are in the same straight line except five points which are collinear.

How many (i) straight lines (ii) triangles can be formed by joining them?

Solution:

(i) Number of straight lines formed joining the 18 points, taking 2 points at a time $={ }^{18} C_{2}=\frac{18}{2} \times \frac{17}{1}=153$

Number of straight lines formed joining the 5 points, taking 2 points at a time $={ }^{5} C_{2}=\frac{5}{2} \times \frac{4}{1}=10$

But, when 5 collinear points are joined pair wise, they give only one line.

$\therefore$ Required number of straight lines $=153-10+1=144$

(ii) Number of triangles formed joining the 18 points, taking 3 points at a time $={ }^{18} C_{3}=\frac{18}{3} \times \frac{17}{2} \times \frac{16}{1}=816$

Number of straight lines formed joining the 5 points, taking 3 points at a time $={ }^{5} C_{3}=\frac{5}{3} \times \frac{4}{2} \times \frac{3}{1}=10$

$\therefore$ Required number of triangles $=816-10=806$