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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!!!

 Jun 2, 2020
 #1
avatar+23 
+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.
  • 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 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!

 Jun 2, 2020
 #2
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+1

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

 Jun 2, 2020
 #3
avatar+31511 
+3

I get 10 different pairs:

 

 Jun 2, 2020
 #4
avatar+80 
+1

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

 Jun 2, 2020
edited by Sarvajit  Jun 2, 2020

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