Question

How many triangles can be obtained by joining 12 points, five of which are collinear ?

Answer 1

Since 5 out of 12 points are collinear, so the number of triangles will be ${}^5{C}_3$ less from ${}^{12}{C}_3$

${=}^{12}{C}_3{-}^5{C}_3$

$=\frac{12!}{3!9!}-\frac{5!}{3!2!}$

$=\frac{12\times 11\times 10}{3\times 2}-\frac{5\times 4}{2}$

$=220-10$

$=210$