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4. Prove that in any party (with at least two people), there are always two people in the party with the same number of friends in that party. (Notice that we say the same number of friends, not the same friends. Also, we assume that if a is friend of b, then also b is friend of a.) (Hint: Distinguish two cases: when there is someone in the party that is friend with everyone and when there is not, and use the pigeonhole principle in each case.)

4. Prove that in any party (with at least two people), there are always two people in the party with the same number of friends in that party. (Notice that we say the same number of friends, not the same friends. Also, we assume that if a is friend of b, then also b is friend of a.) (Hint: Distinguish two cases: when there is someone in the party that is friend with everyone and when there is not, and use the pigeonhole principle in each case.)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem 4: Friend Count in a Party** **Statement:** Prove that in any party (with at least two people), there are always two people in the party with the same number of friends within that party. **Details:** - We focus on having the same number of friends, not the same friends. - The relationship is symmetric: if person *a* is a friend of person *b*, then person *b* is also a friend of person *a*. **Hint for Solution:** Consider two scenarios: 1. There is someone in the party who is friends with everyone. 2. No one in the party is friends with everyone. Utilize the pigeonhole principle for each case to determine why it is always true that two people will have the same number of friends.
Transcribed Image Text:**Problem 4: Friend Count in a Party** **Statement:** Prove that in any party (with at least two people), there are always two people in the party with the same number of friends within that party. **Details:** - We focus on having the same number of friends, not the same friends. - The relationship is symmetric: if person *a* is a friend of person *b*, then person *b* is also a friend of person *a*. **Hint for Solution:** Consider two scenarios: 1. There is someone in the party who is friends with everyone. 2. No one in the party is friends with everyone. Utilize the pigeonhole principle for each case to determine why it is always true that two people will have the same number of friends.
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