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


The answer is 12.

A Former Brilliant Member by A Former Brilliant Member

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1 solution

We can use the handshake formula.

n o . o f h a n d s h a k e = n 2 ( n − 1 ) no.~of~handshake=\dfrac{n}{2}(n-1)

66 = n 2 ( n − 1 ) 66=\dfrac{n}{2}(n-1)

132 = n 2 − n 132=n^2-n

n 2 − n − 132 = 0 n^2-n-132=0

( n + 11 ) ( n − 12 ) = 0 (n+11)(n-12)=0

n = − 11 n=-11 or n = 12 n=12 (reject the negative value)

So n = 12 \boxed{n=12}

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