23 people attend a party. Each person shakes hands with at least one other person. What is the minimum possible number of handshakes?


please explain the answer


I got 253 but that's incorrect (the math was right but the formula was wrong).  

\(x=\text{number of handshakes}\)

\(\frac{23(23-1)}{2}=x \\ \frac{23\times 22}{2}=x \\ x=\colorbox{red}{253}\color{red}{\leftarrow\text{THIS IS WRONG}}\)

lolzforlife  Jul 5, 2018
edited by Guest  Jul 5, 2018
edited by Guest  Jul 5, 2018

the answer is 12


If each person shakes hands with exactly one other person, then there will be 23/2 handshakes, since two people are required to complete a handshake. This is equal to 11.5 handshakes, which is clearly impossible. We can achieve 12 handshakes by forming two rows of 11 and 12 people. Each person in the first row shakes hands a different person in the second row. This will give eleven handshakes. The leftover person must shake hands with someone, giving 12 handshakes total.

lolzforlife  Jul 5, 2018

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