Given a group of 17 people in a party, we consider the number of times a person shakes hand with someone else. Show that there is always two people who shake hands with the same total number of people.

Question

Sophie Chaney

Mathematics

27 June, 12:23

Given a group of 17 people in a party, we consider the number of times a person shakes hand with someone else. Show that there is always two people who shake hands with the same total number of people.

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Jaron Holland · Answer 1

Step-by-step explanation:

Given that there is a group of 17 people in a party. There takes place a number of hand shakes between them.

To show that there is always two people who shake hands with the same total number of people.

If possible let each person did a different number of handshake.

We know that since there are 17 people, there cannot be more than 16 hand shakes for any one in the group.

Hence if different hand shakes must be 0,1,2 ... 16

Consider the last person having 16 hand shakes. He has shaken hand with each other person in the group thus making it clear that no one could have done 0 hand shakes. Since we get a contradiction, our assumption was wrong. That is, there is always two people who shake hands with the same total number of people.