Question

One line forms two regions in a plane. Similarly, two lines in a plane can form a maximum of four regions. These are shown in the figures.
What is the maximum number of regions that can be formed by $4$ lines in a plane ? lines need not be concurrent.

• A. $7$
• B. $8$
• C. $10$
• D. $11$

Answer 1

The correct option is D $11$

Let the no. of regions be a function f(n).

Now

$f\left(1\right)=2$

$f\left(2\right)=4$

$f\left(3\right)=7$

Clearly

$f\left(n\right)-f(n-1)=n$

$f(n-1)-f(n-2)=n-1$

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$f\left(2\right)-f\left(1\right)=2$

Adding we get $f\left(n\right)-f\left(1\right)=2+3+4+5+..............+n-1+n$

$\Rightarrow f\left(n\right)=2+2+3+4+5+...........+n-1+n$

$\Rightarrow f\left(n\right)=1+(1+2+3+4+5+...........+n-1+n)$

$\Rightarrow f\left(n\right)=1+\frac{n(n+1)}{2}$

So $f\left(4\right)=1+10=11$