(I have posted it twice and got no responce. Plz help somebody

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

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

Hope this solutions helps!

It is not 12... but thank you for your time

I get 10 different pairs:

OMG ur such a lifesaver!!!!! You were right it was 10. Thank you for all your time. Thank you fro trying even if you got it wrong. I just want to thank you for all the support yall gave me thanks.

BTW it was me who aksed the question