Question

There are '$p$' points in space of which '$q$' points are coplanar. Then the number of planes formed is

• A. ${}^p{C}_3{-}^q{C}_3$
• B. ${}^p{C}_3{-}^q{C}_3$ + 1
• C. `${}^p{C}_2{-}^q{C}_2$
• D. ${}^p{C}_q{-}^q{C}_2$

Answer 1

The correct option is B ${}^p{C}_3{-}^q{C}_3$ + 1

A plane is formed by taking $3$ points. So if you have $p$ points, then you need to select $3$ points to make a plane.

i.e ${}^P{C}_3$ planes can be formed.

But if $q$ points are coplanar, then they will together form one plane. So we have to subtract this portion from total and will take only one plane for all these points.

Then, number of planes formed are ${}^p{C}_3{-}^q{C}_3+1$.