Question

Let f(n) be the number of regions in which n coplanar circles can divide the plane. If it is known that each pair of circles intersect at two different points and no three of them have common point of intersection, then

• A. f(20) = 382
• B. f(n) is always an even number
• C. $f^{-1}\left(92\right)=10$
• D. f(n) can be odd

Answer 1

Answer: (A), (B), (C)

The correct options are
A f(20) = 382
B f(n) is always an even number
C $f^{-1}\left(92\right)=10$

f(1) = 2
f(n) = f(n - 1) + 2(n - 1); $\forall n\ge 2$
$\Rightarrow$ f(n) = f(n – 1) + 2(n – 1)
f(n) – f(1) = 2(1 + 2 + 3+…..+ n – 1) = (n – 1)n
$\therefore$ f(20) = 382
$n^2-n+2=92\Rightarrow n=10$
as n(n – 1) must be even, f(n) is always even.