Question

If $7$ points out of $12$ points in a space are co-planar, then the number of distinct planes that can be formed will be:

• A. $19$
• B. $158$
• C. $185$
• D. $201$

Answer 1

The correct option is C $185$

There are $7$ co-planar points and $5$ non co-planar. For forming a plane we need $3$ district points, for this there are $3$ possible scenario.

$\left(1\right).$ All $3$ points from the non-coplanar points.

$\left(2\right).$ $1$ point from co-planar, $2$ from non-coplanar.

$\left(3\right).$ $2$ point from co-planar, $1$ from non-coplanar.

$\Rightarrow$ For $\left(1\right)$ Combination $={}^5{C}_3=10.$

$\Rightarrow$ $\left(2\right)$Combination $={}^7{C}_1\times {}^5{C}_2=70.$

$\Rightarrow$ $\left(3\right)$ Combination $={}^7{C}_2\times {}^5{C}_1=105.$

$\Rightarrow$ Total $=10+70+105=185.$

Hence, the answer is $185.$