This question is quite difficult , which can come in olympial or any other compitative exam.

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# At a party, everyone shook hands with everybody else. There were 66 handshakes. How many people were at the party??

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no of handshakes =nC2=n!/(n-2)!(2)!

=n(n-1)(n-2)!/(n-2)!2=n(n-1)/2=

n(n-1)=132=12×11

so

n=no of handshakes =12here we used combinations because in permutations AB is different from BA but in combinations both are same.

Note -:Since if two persons hand shake it means both shake their hands with each other.So here A shakes hand with B is same as B shakes hand with a.