Five people are gathered in a meeting. Some pairs of people shakes hands. An ordered triple of people $(A,B,C)$ is a trio if one of the following is true: A shakes hands with B, and B shakes hands with C, or A doesn't shake hands with B, and B doesn't shake hands with C. If we consider $(A,B,C)$ and $(C,B,A)$ as the same trio, find the minimum possible number of trios.
Problem
Source: 2017 Indonesia MO Problem 2
Tags: combinatorics
07.07.2017 11:02
if any mistake please tell me : treat the five people as five point A,B,C,D,E and consider a triangle ABC for example : color blue segment for handshake , red segment for no handshake , since a triangle has three sides , there must be two segment has the same color and thus a trio formed. (if a triangle has all three sides the same color , three trios formed and this case should be avoided) for a five points graph , there exist C(5,3) = 10 different triangles , so the minimum posibble number of trios is 10. attached graph is an example pattern for 10 trios.
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01.05.2022 21:14
@above I don't believe your example is right, since although the problem said $(A,B,C)$ is equivalent to $(C,B,A), $ that doesn't necessarily mean that it is also equivalent to $(B,C,A).$ Take the top three vertices for example. Your solution states that it counts as one trio, when in reality taking the permutations results in three trios.