Question

PLEASE HELP I'M DYING INSIDE! Six people are sitting around a circular table, and each person has either blue eyes or green eyes. Let x be the number of people sitting next to at least one blue-eyed person, and let y 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.)

Answer 1

10

Explanation:

The attached figure shows all the cases and counts for at least as many blue as green. The resulting ordered pairs are ...

(6, 0), (6, 2), (6, 4), (5, 3), (5, 5), (3, 3)

For at least as many green as blue, the numbers are reversed. That means there are a total of 10 possible ordered pairs:

(6, 0), (6, 2), (6, 4), (5, 3), (5, 5), (3, 3), (3, 5), (4, 6), (2, 6), (0, 6)