6 people are sitting around a table. Let x be the number of people sitting next to at least one woman and y be the number of people sitting next to at least one man. How many possible values of the ordered pair (x,y) are there? (For example, (6,0) is the pair if all 6 people are women, since all 6 people are sitting next to a woman, and 0 people are sitting next to a man.)
6 people
x be the number of people sitting next to at least one woman
y be the number of people sitting next to at least one man
I tried to do some of this with combinations but it didn't work.
I ended up drawing lots of hexagons, and just looking at what happens.
I think it is correct. 
(6,0) if all are women
(5,2) if 1 man
(4,4) if 2 men sitting together
(6,3) if the 2 men and they are separated by 1 seat
(6,4) if the 2 men are opposite each other
This pattern must be symmetrical
(0,6) if all are men
(2,5) if 1 woman
(4,4) if 2 woman and they sit together
(3,6) if the 2 mwomen and they are separated by 1 seat
(4,6) if the 2 women are opposite each other
This one was harder. But after drawing pics it seems to me that there are only 3 posibilities for 3 men and 3 women.
(5,5) The men all sit together
(5,5) A pair of men, a pair of women, a man, a woman
(6,6) Alternating
so what do we have
(6,0), (0,6)
(5,2)(2,5)
(4,4)
(6,3)(3,6)
(6,4)(4,6)
(5,5)
(6,6)
11 pairs