23 people attend a party. Each person shakes hands with at most 22 other people. What is the maximum possible number of handshakes, assuming that any two people can shake hands at most once?

tertre
Feb 18, 2018

#1**+2 **

Notice that the 23rd person can shake hands with 22 other people

And the 22nd person can shake hande with 21 people [ this person has already shaken hands with the 23rd person]

And the 21st person can shake hands with 20 people[ this person has already shaken hands with the first two ]

So....the total possible handshakes are

22 + 21 + 20 + ......+ 1 =

1 + 2 + 3 + ......+ 20 + 21 + 22

Which is just the sum of the first 22 positive integers and is given by :

(22)(23) / 2 = 11 * 23 = 253 handshakes

CPhill
Feb 18, 2018