Question

There are $p$ points in space, no four of which are in the same plane with the exception of $q$, which are all in the same plane; find how many planes there are each containing three of the points.

Answer 1

Planes are to be formed in space with the help of $3$ points.
Planes to be formed out of ${}^{\prime }{p}^{\prime }$ points $=$ ${}^p{C}_3$
Planes to be formed out of ${}^{\prime }{q}^{\prime }$ points $=$ ${}^q{C}_3$
But ${}^{\prime }{q}^{\prime }$ points are in single plane. Hence, these points cannot form a distinct plane.
$\therefore$ Total Number of planes $={}^p{C}_3-{}^q{C}_3+1$
$\therefore$ Total no. of planes $=\frac{p(p-1)(p-2)}{6}-\frac{q(q-1)(q-2)}{6}+1$