# The number of triangles that are formed by choosing the vertices from a set of 12 points,seven of which lie on the same line is

$\begin{array}{1 1}(A)\;105\\(B)\;15\\(C)\;175\\(D)\;185\end{array}$

Toolbox:
• $C(n,r)=\large\frac{n!}{r!(n-r)!}$
Total no of points =12
No of points in same line =7
Number of triangles formed =$12C_3-7C_3$
$12C_3=\large\frac{12!}{3!9!}$
$\Rightarrow \large\frac{12\times 11\times 10\times 9!}{3\times 2\times 9!}$$=220$
$7C_3=\large\frac{7!}{3!4!}$
$\Rightarrow \large\frac{7\times 6\times 5\times 4!}{3\times 2\times 4!}$
$\Rightarrow 35$
$12C_3-7C_3=220-35$
$\Rightarrow 185$
Hence (D) is the correct answer