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

The correct option is C $185$

\therefore A triangle can be formed by joining 3 points.

There are a total of 12 points, so 3 points can be selected from 12 points in ${}^{12}{C}_3$ ways.

Now it is given that 7 points lie in the same straight line, so joining these points will not give any triangle.

But these are also involved in ${}^{12}{C}_3$ ways.

Therefore we have to subtract ${}^7{C}_3$ from ${}^{12}{C}_3$ to get the required triangle.

$\therefore$ Required triangle = ${}^{12}{C}_3{-}^7{C}_3$

$=220-35$

$=185$

$\therefore$ so, the answer is C.185