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.)
+118608 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
+118608 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
There can be:
1. 6 women, no men: (6, 0) (only possible way) (all sitting next to a woman)
2. 5 women, 1 man: (6, 2) (only possible way) (all 6 must sit next to a woman, 2 woman sitting next to the man)
3. 4 women, 2 men: (6, 4) (if the men sit next to or opposite of each other); or (5, 3) (if they sit one apart)
4. 3 women, 3 men: Arrangements can be WWWMMM or WWMWMM or MMWMWW, so it's (5, 5); or WMWMWM would be (3, 3)
5. 2 women, 4 men: Symetrical with 3., (4, 6) or (3, 5)
6. 1 woman, 5 men: Symetrical with 2., (2, 6)
7. 0 women, 6 men: Symmetrical with 1., (0, 6)
That's all!
Case 1: 6 women. Then (6,0) is the only possible ordered pair.
Case 2: 5 women and 1 man. Then (6,2) is the only possible ordered pair: all 6 people must be sitting next to a woman, and 2 of the women are sitting next to the man.
Case 3: 4 women and 2 men. Then, if the men sit next to each other or opposite each other, our ordered pair is (6,4); if they sit one apart, our ordered pair is (5,3).
Case 4: 3 women and 3 men. Then if the arrangement is WWWMMM, WWMWMM, or MMWMWW, our ordered pair is (5,5). If it is WMWMWM, our ordered pair is (3,3).
Case 5: 2 women and 4 men. By symmetry with Case 3, the possible pairs are (4,6) and (3,5).
Case 6: 1 woman and 5 men. By symmetry with Case 2, the only possible pair is (2,6).
Case 7: 6 men. (0,6)
This yields a total of 10 ordered pairs.
