An elementary school teacher has a small class consisting of 6 students. 9a. When handing out name tags at random, what is the probability that at least 1 classmate gets their name tag? 9b. Some of the classmates might already be friends. Some of them might be strangers. Assuming that "friend" is a symmetric relationship (if Johnny is friends with Robert, then Robert is also friends with Johnny), show there must exist at least 3 of these students who are all friends or at least 2 who are all strangers. If 1 of the 6 students drops the class, must this still be true? Explain.
9. An elementary school teacher has a small class consisting of 6 students.
9a. When handing out name tags at random, what is the
9b. Some of the classmates might already be friends. Some of them might be strangers. Assuming that "friend" is a symmetric relationship (if Johnny is friends with Robert, then Robert is also friends with Johnny), show there must exist at least 3 of these students who are all friends or at least 2 who are all strangers. If 1 of the 6 students drops the class, must this still be true? Explain.
By intuition, I would think there is a 1/6 chance a student will get their name tag. Regarding the second part of the question, by intuition I would also think that the given conditions would still be true if 1 student drops the class. We have enough of a sample size to not worry about removing 1 student. Is my intuition correct? If not, why?
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Is there anyway to show this is true via Pigeonhole Principle?
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