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Combination Based on Geometry

If there are ...

Question

If there are $n$ circles in a plane which gives $56$ points of intersection, then the minimum value of $n$ is

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Solution

The maximum number of points of intersection of $n$ circles in a plane is $2 \times^{n} C_{2}$
$∴$ min value of $n$ will be
$2 \times^{n} C_{2} = 56$
$\Rightarrow 2 \times \frac{n (n - 1)}{2} = 56$
$\Rightarrow n^{2} - n - 56 = 0$
$\Rightarrow (n - 8) (n + 7) = 0$
$\Rightarrow n = 8$ (min)

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Combination Based on Geometry

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