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.)
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The number of ordered pairs (x,y) is 22.
To determine the number of possible values for the ordered pair (x, y), we need to consider the potential arrangements of men and women around the table.
Let us start by finding out the possible values for x. To have x people sitting next to at least one woman, there must be at least one woman seated in each group. There are three groups where this could occur:
1) One woman is seated with five men: In this case, one man will be sitting next to the woman, so x = 1.
2) Two women are seated next to each other with four men: In this case, two men will be sitting next to the women, so x = 2.
3) Three women are seated next to each other with three men: In this case, three men will be sitting next to the women, so x = 3.
Now, let's consider the possible values for y. Since y represents the number of people sitting next to at least one man, it depends on the arrangement of women around the table.
1) Three women are seated next to each other: In this case, no men will be sitting next to women, so y = 0.
2) Two women are seated next to each other: Here, there will be one man sitting next to one woman, so y = 1.
3) One woman is seated next to another woman: In this case, two men will be sitting next to the women, so y = 2.
4) Women are spread out, with no adjacent women: If the women are evenly spread out, each woman will have two men sitting next to her, so y = 2.
Applying the possible values of x and y, we can create the following ordered pairs:
(1,0), (1,1), (2,0), (2,1), (3,0), (3,2)
Therefore, there are six possible values for the ordered pair (x, y).