Question

If 7 points out of 12 are in the same straight line, then the number of triangles formed is
(A) 19
(B) 158
(C) 185
(D) 201

Answer 1

First of all, find the ways formed triangles by 12 points that is ${}^{12}{C_3}$ then after 7 points are in a straight line so find the ways formed triangles by 7 points that is ${}^7{C_3}$then subtract this from ${}^{12}{C_3}$ so, we get the answer.
The formula to calculate ${}^n{C_r}$ is:
${}^n{C_r} = \left[ {\dfrac{{n!}}{{r!(n - r)!}}} \right]$

Complete step by step solution:
To formed a triangle, we need to three points so if 12 points are formed ${}^{12}{C_3}$ triangles.
Total triangle formed by 12 points $= {}^{12}{C_3}$
By using this formula ${}^n{C_r} = \left[ {\dfrac{{n!}}{{r!(n - r)!}}} \right]$we can expand ${}^{12}{C_3}$like this
$= \dfrac{{12!}}{{(3!)(12 - 3)!}}$
In above equation 12! Expand by using formula $n! = n \times (n - 1) \times (n - 2) \times (n - 3) \times ........1$
$= \dfrac{{12 \times 11 \times 10 \times 9!}}{{(3 \times 2 \times 1)(9!)}}$
$= 220$triangles
But in a statement that clearly stated as that seven points are in the same line that seven points formed triangle ${}^7{C_3}$ways so that seven points not formed a triangle with each other so subtract this way of the Striangle from the total ways of the triangle.
So, If the 7 points out of 12 are $= {}^{12}{C_3} - {}^7{C_3}$
By using this formula ${}^n{C_r} = \left[ {\dfrac{{n!}}{{r!(n - r)!}}} \right]$we can expand ${}^7{C_3}$like this
$= 220 - \left[ {\dfrac{{7!}}{{3!(7 - 3)!}}} \right]$
$= 220 - 35$
$= 185$ triangles

$\therefore$ The number of triangles formed is 185.

Note:
If there is $x$ point to form a triangle the ways of the triangle to form are ${}^x{C_3}$. But not to forget if any points like p point are in a straight line then this point form ${}^p{C_3}$ triangle not to forgot to subtract this triangle.