+0

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

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2136
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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.)

Apr 1, 2015

#1
+95334
+4

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

Apr 2, 2015

#1
+95334
+4

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

Melody Apr 2, 2015
#2
+1

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!

Nov 13, 2015
#3
+2

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.

Apr 10, 2016