Six people are sitting around a circular table, and each person has either blue eyes or green eyes. Let be the number of people sitting next to at least one blue-eyed person, and let be the number of people sitting next to at least one green-eyed person. How many possible ordered pairs (\((x, y)\) are there? (For example, \((x,y) = (6,0)\) if all six people have blue eyes, since all six people are sitting next to a blue-eyed person, and zero people are sitting next to a green-eyed person.)
I'm somewhat confused on how to approach this question, could someone please guide me through it?
I counted them and got
6 of 1 and 0 of the other (6,0) (0,6)
5 of 1 and 1 to the other (6,2) (2,6)
4 of 1 and 2 of the other (6,3) (3,6) also (6,2) (2,6) again
3 each (5,5)
So I counted 14 possibilities