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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 in two different points and no three of them have common point of intersection, then 


A
f(15)=212
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B
f1(134)=12  
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C
f(n) is always an even number 
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D
f(n) is a perfect square 
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Solution

The correct options are
A f(15)=212
B f1(134)=12  
C f(n) is always an even number 
Let the number of regions for n circles be f(n).
f(1)=2
Now,
f(n)=f(n1)+2(n1),   n2
f(n)f(n1)=2(n1)
Putting n=2,3,...,n, we get
f(n)f(1)f(n)f(1)=2(1+2+3+...+n1)f(n)f(1)=(n1)n
f(n)=n(n1)+2
f(n)=n2n+2
f(n) is always even.

f(15)=212

n2n+2=134
n2n132=0
n=12 (since, n can't be negative)
 

Mathematics

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