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.) I got 14 and 8 but it said it was wrong. I need help

Please this is my 3rd time asking this question. It would mean a lot. Thank you for your time!!!

Guest Jun 2, 2020

#1**+1 **

You can just brute force this one:

6 cases:

- all 6 blue
- this case is pretty easy to see, (6,0).
**1**solution for this case.

- this case is pretty easy to see, (6,0).
- 5 blue 1 green
- two people sit next to the person with green eyes and everyone is sitting next to someone with blue eyes (6, 2).
**1**solution for this case as well.

- two people sit next to the person with green eyes and everyone is sitting next to someone with blue eyes (6, 2).
- 4 blue 2 green
- if the greens are next to each other, there is one solution (6, 2)
- if the greens are one apart, there is another solution (6, 3)
- if the greens are two apart, there is one more (6, 4), for
**3**solutions in this case.

- 3 blue 3 green
- all 3 greens next to eachother (5, 5)
- 1 green separate from the other 2 (5,5)
- blue and greens alternate (3,3). In this case we have a repeating solution so only
**2**more posibilities.

- 2 blue 4 green
- same as the 4 blue 2 green, just flipped for
**3**more solutions.

- same as the 4 blue 2 green, just flipped for
- 1 blue 5 green
- same as the 5 blue 1 green, just flipped for
**1**additional ordered pair.

- same as the 5 blue 1 green, just flipped for
- all 6 green
- same as all 6 blue, just flipped for
**1**additional ordered pair.

- same as all 6 blue, just flipped for

Adding up the bolded numbers, we get a final answer of *12* ordered pairs.

Hope this solutions helps!

North Jun 2, 2020