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

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Re: In how many ways can 3 men and 3 women be seated around a [#permalink]
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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Re: In how many ways can 3 men and 3 women be seated around a [#permalink]
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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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Re: In how many ways can 3 men and 3 women be seated around a [#permalink]
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OA is 12. But I dont seem to get it.
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Re: In how many ways can 3 men and 3 women be seated around a [#permalink]
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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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Re: In how many ways can 3 men and 3 women be seated around a [#permalink]
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.
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Re: In how many ways can 3 men and 3 women be seated around a [#permalink]
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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Re: In how many ways can 3 men and 3 women be seated around a [#permalink]
[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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Re: In how many ways can 3 men and 3 women be seated around a [#permalink]
I accept...
Gayathri is right should be 72/6 = 12, forgot that its circular
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Re: In how many ways can 3 men and 3 women be seated around a [#permalink]
circular permutation link is helpful.. do we see such questions on GMAT?
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Re: In how many ways can 3 men and 3 women be seated around a [#permalink]
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 ..
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Re: In how many ways can 3 men and 3 women be seated around a [#permalink]
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The answer GMAT wants is 12. You fix one guy to a seat and alternate man-woman. Now, the other two guys can be arranged in only two different ways with respect to that one guy you fixed in one seat. So, that's 2! ways. The women can be arranged in 3! ways with respect to the one guy. So, it's 2!*3!.

However

If each chair itself is unique, then shouldn't the answer be 72? The question assumes that the seats themselves don't matter, and that only the relative arrangement of people matters. So, if the seats actually mattered, say, seat A is a terrible chair that's about to fall apart, seat B is an Eames lounge chair, seat C is a massage chair, seat D is closest to the window, seat E is closest to the bathroom, and seat F is next to a different loud table, then such a scenario does exist, and the answer would need to be 72.

So, calculate 12 as you did above from one fixed guy in seat A, then move the guy you fixed to seat B, then C, then D, etc. That's 72 combinations, since each seat with respect to one guy has 12 arrangements and there are six seats total (12*6). Another way to do it is to think of each seat as M1-W2-M3-W4-M5-W6 (man seat#; woman seat#) around a table and multiply the combination, so after simplyfing it's 3*3*2*2*1*1 = 36. Those are only for the odd numbered seats for men, and even numbered seats for women, so the answer needs to be double that, giving us 72.

If 12 and 72 were both answer choices, I'd pick 12 but with serious protest. GMAT may ask for unique orders of people only, so the same order but in different chairs may be disqualified in GMAT's eyes. But, the GMAT doesn't specify that they want only unique orders of people - is it somehow implied in the question? The question itself is insufficient to draw a conclusion, esp. if 12 and 72 are both answer choices. Any thoughts?
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Re: In how many ways can 3 men and 3 women be seated around a [#permalink]
first solve the question without constraint given .then it would be like this .. 3!(3-1)!=12.
hence there are 12 ways .what can we infer from this is that the answer will be definately less than 12 if we add the constraint.
so mens places are fixed here.as the arrangement is round so there will be (3-1)! ways =2 ways
after this women position should be between two men.so here also there will be three places remaining .as the arrangement is round there will be (3-1)! ways=2 ways. therefore total of 4 ways..
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Re: In how many ways can 3 men and 3 women be seated around a [#permalink]
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iPen wrote:
The answer GMAT wants is 12. You fix one guy to a seat and alternate man-woman. Now, the other two guys can be arranged in only two different ways with respect to that one guy you fixed in one seat. So, that's 2! ways. The women can be arranged in 3! ways with respect to the one guy. So, it's 2!*3!.

However

If each chair itself is unique, then shouldn't the answer be 72? The question assumes that the seats themselves don't matter, and that only the relative arrangement of people matters. So, if the seats actually mattered, say, seat A is a terrible chair that's about to fall apart, seat B is an Eames lounge chair, seat C is a massage chair, seat D is closest to the window, seat E is closest to the bathroom, and seat F is next to a different loud table, then such a scenario does exist, and the answer would need to be 72.

So, calculate 12 as you did above from one fixed guy in seat A, then move the guy you fixed to seat B, then C, then D, etc. That's 72 combinations, since each seat with respect to one guy has 12 arrangements and there are six seats total (12*6). Another way to do it is to think of each seat as M1-W2-M3-W4-M5-W6 (man seat#; woman seat#) around a table and multiply the combination, so after simplyfing it's 3*3*2*2*1*1 = 36. Those are only for the odd numbered seats for men, and even numbered seats for women, so the answer needs to be double that, giving us 72.

If 12 and 72 were both answer choices, I'd pick 12 but with serious protest. GMAT may ask for unique orders of people only, so the same order but in different chairs may be disqualified in GMAT's eyes. But, the GMAT doesn't specify that they want only unique orders of people - is it somehow implied in the question? The question itself is insufficient to draw a conclusion, esp. if 12 and 72 are both answer choices. Any thoughts?

You are absolutely correct about both your calculations however there here is a concrete point to know about GMAT

You don't consider Chairs different until GMAT clearly mentions it and No mention of the same would be considered that the arrangement takes into account only the positions at which people sit and not the chairs on which people sit
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Re: In how many ways can 3 men and 3 women be seated around a [#permalink]
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GMATinsight wrote:
iPen wrote:
The answer GMAT wants is 12. You fix one guy to a seat and alternate man-woman. Now, the other two guys can be arranged in only two different ways with respect to that one guy you fixed in one seat. So, that's 2! ways. The women can be arranged in 3! ways with respect to the one guy. So, it's 2!*3!.

However

If each chair itself is unique, then shouldn't the answer be 72? The question assumes that the seats themselves don't matter, and that only the relative arrangement of people matters. So, if the seats actually mattered, say, seat A is a terrible chair that's about to fall apart, seat B is an Eames lounge chair, seat C is a massage chair, seat D is closest to the window, seat E is closest to the bathroom, and seat F is next to a different loud table, then such a scenario does exist, and the answer would need to be 72.

So, calculate 12 as you did above from one fixed guy in seat A, then move the guy you fixed to seat B, then C, then D, etc. That's 72 combinations, since each seat with respect to one guy has 12 arrangements and there are six seats total (12*6). Another way to do it is to think of each seat as M1-W2-M3-W4-M5-W6 (man seat#; woman seat#) around a table and multiply the combination, so after simplyfing it's 3*3*2*2*1*1 = 36. Those are only for the odd numbered seats for men, and even numbered seats for women, so the answer needs to be double that, giving us 72.

If 12 and 72 were both answer choices, I'd pick 12 but with serious protest. GMAT may ask for unique orders of people only, so the same order but in different chairs may be disqualified in GMAT's eyes. But, the GMAT doesn't specify that they want only unique orders of people - is it somehow implied in the question? The question itself is insufficient to draw a conclusion, esp. if 12 and 72 are both answer choices. Any thoughts?

You are absolutely correct about both your calculations however there here is a concrete point to know about GMAT

You don't consider Chairs different until GMAT clearly mentions it and No mention of the same would be considered that the arrangement takes into account only the positions at which people sit and not the chairs on which people sit

Gotcha. I'll stick with GMAT's preferences.
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Re: In how many ways can 3 men and 3 women be seated around a [#permalink]
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Deal with men and women separately:

Seating Layout: W M W M W M

Because they are in a circle, women can sit in (n-1)! ways as you can imagine them all getting up and moving one seat to the side and they would be in the same order.

Now that the women are in place the men need to be dealt with as if their positions were fixed because they are no longer sitting relative only to each other, but now are sitting relative to the women. The men can sit in n! ways.

Together you have W! x M! or 2x1 x 3x2x1 = 12
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Re: In how many ways can 3 men and 3 women be seated around a [#permalink]
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I have one doubt if we are fixing one man's seat that is we are going to select that one man from the 3 man's available then it should be 3 why we are choosing 1?

Pls clear my doubt
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Re: In how many ways can 3 men and 3 women be seated around a [#permalink]
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Gnanam wrote:
I have one doubt if we are fixing one man's seat that is we are going to select that one man from the 3 man's available then it should be 3 why we are choosing 1?

Pls clear my doubt

Please have a look at the following link. It explains in detail why only 1 should be considered

Coming to the question (however, I cannot show circular arrangement.. to make it simple, i have taken straight arrangemnet) -

seat1 seat2 seat3 seat4 seat5 seat6
---- ---- ----- ----- ----- -----

Seat1 - Man - 1 way (because, he can choose any of seats with circular arrangement)
Seat2 - should be women (from question) - can be done in 3 ways (because women can be seated only in seat-2,4 & 6)
Seat3 - should be Man - can be done in 2ways (again, only seat 3 & 5 are available)
Seat4 - should be woman - can be seated in 2 ways (seat 4 & 6 are available)
seat5 - should be man - only 1 way (no more seats for man left)
seat6 -should be woman - only 1 way (no more seats left)

Total arrangements = 1 x 3 x 2 x 2 x1 x1 = 12 ways
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