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? (For example, (a,b) = (6,0) if all six people are tall, since all six people are sitting next to a tall person, and zero people are sitting next to a short person.)
6 people
x be the number of people sitting next to at least one tall person
y be the number of people sitting next to at least one short person
(6,0) if all are short
(5,2) if 1 tall person
(4,4) if 2 tall people sitting together
(6,3) if 2 tall people and they are separated by 1 seat
(6,4) if 2 tall people are opposite each other
This pattern must be symmetrical
(0,6) if all are tall people
(2,5) if 1 short person
(4,4) if 2 short person and they sit together
(3,6) if 2 short people and they are separated by 1 seat
(4,6) if the 2 short people are opposite each other
This one was harder. But after drawing pics it seems to me that there are only 3 possibilities for 3 tall people and 3 short people.
(5,5) The tall people all sit together
(5,5) A pair of tall people, a pair of short people, a tall person, a short person
(6,6) Alternating
so what do we have
(6,0), (0,6)
(5,2)(2,5)
(4,4)
(6,3)(3,6)
(6,4)(4,6)
(5,5)
(6,6)
11 pairs
