Question

please solve 22.1, using the Theorem given. Thank you.

Theorem 22.1. Suppose that n people (n 2 2) are at a party. Then there exist at least two people at the party who know the same number of people present First you need to know the rules. We will assume that no one knows him- or herself. We will also assume that if x claims to know y, then y also knows x. The idea behind the proof is this, and you can try it out at your next party. You will put n boxes on the board numbered 0 through n -1. Each person counts up the number of people he or she knows at the party. You ask them that number and write their name in the box with the same number. Note that each person's answer corresponds to exactly one of the boxes 0 through n -1. The theorem claims that at least two people's names will end up in the same box. Proof. We imagine n boxes that are numbered 0 through n-1. For an integer m with 0Smsn-1, box m contains the names of the people who know m people at the party We break this proof into two cases. First, suppose that there is someone at the party who doesn't know anyone. We'll call this party crasher Ms. X. Now if we piclk know himself and, since Ms. X doesn't know some other party attendee, he doesn't Problem 22.1. Consider the story of n people at a party in Theorem someone else has a rival party the same evening, and no one can attend both. 22.1. Suppose Som party and shows it to everyone at your aVour party isn't that much fun, so you each look at the picture and say how ople you know at the other party. No one says the same number. What can one takes a picture of the people at the rival y. vou conclude about the number of people attending the other party?

Answer 1

5040 - 2lo 84