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  4. Everybody in a room shakes hands with everybody else. The total number of handshakes is 66 . The number of persons in the room is

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Q. Everybody in a room shakes hands with everybody else. The total number of handshakes is $66$ . The number of persons in the room is

ManipalManipal 2018

Solution:

Let the number of persons in the room be $n$, then
${ }^{n} C_{2}=66 $
$\Rightarrow \frac{n(n-1)}{2}=66 $
$\Rightarrow n^{2}-n-132=0$
$\Rightarrow(n-12(n+11)=0 $
$\Rightarrow n=12$