Prove that any group of 20 people will contain at least one pair of people with the same amount of friends within the group. (Here, you can let S = {p1, P2, ..., P20} be an arbitrary set of 20 people, and define n(pi) for 1 < i < 20 to be the number of friends for person pi within this group. Assume friendship is symmetric, so if someone has 0 friends in the group then there can't be someone with 19 friends. Similarly, if someone has 19 friends in the group then there can't be anyone with 0 friends in the group. What are the possible values of n(p;).?).
Prove that any group of 20 people will contain at least one pair of people with the same amount of friends within the group. (Here, you can let S = {p1, P2, ..., P20} be an arbitrary set of 20 people, and define n(pi) for 1 < i < 20 to be the number of friends for person pi within this group. Assume friendship is symmetric, so if someone has 0 friends in the group then there can't be someone with 19 friends. Similarly, if someone has 19 friends in the group then there can't be anyone with 0 friends in the group. What are the possible values of n(p;).?).
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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Transcribed Image Text:Prove that any group of 20 people will contain at least one pair of people with the same amount of friends within the group. (Here, you can let \( S = \{ p_1, p_2, \ldots, p_{20} \} \) be an arbitrary set of 20 people, and define \( n(p_i) \) for \( 1 \leq i \leq 20 \) to be the number of friends for person \( p_i \) within this group. Assume friendship is symmetric, so if someone has 0 friends in the group then there can't be someone with 19 friends. Similarly, if someone has 19 friends in the group then there can't be anyone with 0 friends in the group. What are the possible values of \( n(p_i) \)?)
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