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.)
+379 little bit late :)
the answer is 10
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)
