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?

Answer 1

(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.
∴ 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 = ${}^5{C}_3=\frac{5}{3}\times \frac{4}{2}\times \frac{3}{1}=10$
∴ Required number of triangles = $816-10=806$