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In how many ways can 3 men and 3 women be seated around a

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In how many ways can 3 men and 3 women be seated around a [#permalink] New post 05 Jan 2005, 07:41
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In how many ways can 3 men and 3 women be seated around a round table if each women is to be between 2 men.

Could somebody explain.

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 [#permalink] New post 05 Jan 2005, 08:04
72 ways.

In a round table the first person can be selected in 6 ways (any men or women can be selected), next one can be selected in 3 ways (i.e. opposite gender of the person selected in the first chair), next set can be taken by 2 remaining from an opposite gender from the 2nd chair, so 2 ways, 4th seat again can be filled by 2 remaining from opposite gender from the 3rd chair, last 2 chairs can be filled 1 way each. So no of ways:

6*3*2*2*1*1 = 72 ways. Is that correct ?
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 [#permalink] New post 05 Jan 2005, 09:01
You can do this problem in number of ways:

Please draw a circle and mark 6 points on it. 3 x and 3 .
1)
a) split into 3 men and 3 women.
b) take men first -> first person can be seated in any of the available seats so # of ways -> 6
c) 2 men are remaining and only 2 seats are available. Since one man is fixed, adjacent positions cannot be occupied by men, also seat exactly opposite to first man cannot be occupied by any man. Thus we have only 2 places.
d) # of ways 2nd man can sit -> 2
e) last man has only one way.
f) 3 women are remaining and 3 places. 3 out of six have been occupied by 3 men.
g) First women can be placed in 3 ways
h] second women in 2 way and
i] third women in 1 way.

so combining 6*2*3*2 -> 72
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 [#permalink] New post 05 Jan 2005, 12:04
OA is 12. But I dont seem to get it.
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 [#permalink] New post 05 Jan 2005, 12:43
Three men can be seated first at the round table in 2! = 2 ways.
Then the three women can be seated in 3 gaps in 3! = 6 ways.
Hence the required number of ways = 2 x 6 = 12
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 [#permalink] New post 05 Jan 2005, 13:06
I cannot give you an answer in terms of a equation. What I can do is explain it in a diagram. So if you draw a circle and draw six lines aorund that circle representing the six places where the people sit and then put man 1(m1), m2 and m3 in their place and the women between them. If you rotate the men (3 options) and you rotate the women (3 options) around the table you get a total 6 options. If you then switch the men and women and do the same you get another 6 and add these together you get 12 posibilities. I know it is crude but it worked for me. :-D
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 [#permalink] New post 05 Jan 2005, 13:42
gayathri wrote:
Three men can be seated first at the round table in 2! = 2 ways.
Then the three women can be seated in 3 gaps in 3! = 6 ways.
Hence the required number of ways = 2 x 6 = 12


Can you explain why the ways the men can be seated is 2! and not 3!
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All - I thought it would be a good idea to put together [#permalink] New post 05 Jan 2005, 14:57
toddmartin wrote:
Can you explain why the ways the men can be seated is 2! and not 3!


Since I cannot attach any figures, refer to this URLfor explanation of circular permuations. This is my explanation...

1-2-3-4-5-6-1 (assume this to be a closed circle)

Lets assume the position of the first man to be "fixed" at 1, then there are two remaining positions for the remaining two men: 3rd and 5th seat.
Number of ways of seating a man in 3rd seat = 2
As only one man is left, number of ways of seating a man in 5th seat is 1.
Number of ways that three men can be seated first at the round table = 1*2*1=2

There are three remaining seats 2,4, and 6.
Number of options for 2 = 3
Number of options for 4 = 2 (only 2 women are left)
Number of options for 6 = 1 (only 1 woman left)
i.e. The three women can be seated in 3! ways = 6

Total number of possibilities = 2*6 = 12
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 [#permalink] New post 05 Jan 2005, 18:35
[quote="banerjeea_98"]72 ways.

In a round table the first person can be selected in 6 ways (any men or women can be selected), next one can be selected in 3 ways (i.e. opposite gender of the person selected in the first chair), next set can be taken by 2 remaining from an opposite gender from the 2nd chair, so 2 ways, 4th seat again can be filled by 2 remaining from opposite gender from the 3rd chair, last 2 chairs can be filled 1 way each. So no of ways:

6*3*2*2*1*1 = 72 ways. Is that correct ?[/quote]

Thx gayathri for the article, forgot abt circular prob, ofcourse, 72 needs to be divided by 6 ppl = 72/6 ==> 12 ways.
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 [#permalink] New post 06 Jan 2005, 05:18
I accept...
Gayathri is right should be 72/6 = 12, forgot that its circular
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 [#permalink] New post 06 Jan 2005, 06:47
circular permutation link is helpful.. do we see such questions on GMAT?
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 [#permalink] New post 06 Jan 2005, 16:34
Another way of looking at..
The 1st person has 1 position only,not 6..On a circular table all the seats r the same..It doesn't make a lot of sense to get to 72 and then divide it by 6! Or so i think..
Good luck to all :!: ..
  [#permalink] 06 Jan 2005, 16:34
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