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# Casework question

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There are six people sitting at a circular table. Each person is either tall or short. Let a be the number of people sitting next to at least one tall person, and let b be the number of people sitting next to at least one short person. How many possible ordered pairs (a,b) are there?

May 1, 2020

#1
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Using casework

Case 1: 6 tall people. Then (6,0) is the only possible ordered pair.

Case 2: 5 tall people and 1 short person. Then (6,2) is the only possible ordered pair: all 6 people must be sitting next to a tall person, and 2 of the tall people are sitting next to the short person.

Case 3: 4 tall people and 2 short people. Then, if the short people sit next to each other or opposite each other, our ordered pair is (6,4); if they sit one apart, our ordered pair is (5,3).

Case 4: 3 tall people and 3 short people. Then if the arrangement is SSSTTT, SSTSTT, or TTSTSS, our ordered pair is (5,5). If it is STSTST, our ordered pair is (3,3).

Case 5: 2 tall people and 4 short people. By symmetry with Case 3, the possible pairs are (4,6) and (3,5).

Case 6: 1 tall person and 5 short people. By symmetry with Case 2, the only possible pair is (2,6).

Case 7: 6 short people. The only possible pair is (0,6).

This gives us a total of 10 ordered pairs.

May 1, 2020
edited by LuckyDucky  May 2, 2020

#1
+651
+2

Using casework

Case 1: 6 tall people. Then (6,0) is the only possible ordered pair.

Case 2: 5 tall people and 1 short person. Then (6,2) is the only possible ordered pair: all 6 people must be sitting next to a tall person, and 2 of the tall people are sitting next to the short person.

Case 3: 4 tall people and 2 short people. Then, if the short people sit next to each other or opposite each other, our ordered pair is (6,4); if they sit one apart, our ordered pair is (5,3).

Case 4: 3 tall people and 3 short people. Then if the arrangement is SSSTTT, SSTSTT, or TTSTSS, our ordered pair is (5,5). If it is STSTST, our ordered pair is (3,3).

Case 5: 2 tall people and 4 short people. By symmetry with Case 3, the possible pairs are (4,6) and (3,5).

Case 6: 1 tall person and 5 short people. By symmetry with Case 2, the only possible pair is (2,6).

Case 7: 6 short people. The only possible pair is (0,6).

This gives us a total of 10 ordered pairs.

LuckyDucky May 1, 2020
edited by LuckyDucky  May 2, 2020
#2
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You copied the AoPS solution.

May 3, 2020