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Seven men and five women have to sit around a circular table

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Re: Seven men and five women have to sit around a circular table  [#permalink]

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New post 18 May 2014, 04:53
Awesome approach Bunuel...
Why is this wrong neways: 12 Guys..No of ways 11!
Take 2 Women and bind them together...Now 11 guys and no of arrangements is 10!*2!
This covers all arrangements with at least 2 women together....

No of arrangements for no women together is: 11!- 10!*2! ---> wrong answer
Why?
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Re: Seven men and five women have to sit around a circular table  [#permalink]

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New post 18 May 2014, 06:10
M-M-M-M-M-M-M-

Bold M is fixed at top of circular arrangement
Women can occupy any of the 7 "-"

Men permutations = 6!
Women combos × permutations 7C5 × 5!

Total arrangements 6! 5! (21)
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Re: Seven men and five women have to sit around a circular table  [#permalink]

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New post 18 May 2014, 08:03
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JusTLucK04 wrote:
Awesome approach Bunuel...
Why is this wrong neways: 12 Guys..No of ways 11!
Take 2 Women and bind them together...Now 11 guys and no of arrangements is 10!*2!
This covers all arrangements with at least 2 women together....

No of arrangements for no women together is: 11!- 10!*2! ---> wrong answer
Why?



The error is in 10!*2!. This is not taking into account which 2 women are together. This is always done when you know that which two persons are together. For example, if a question says that A & B should always be together, then only you can do it. Now they are not specifying any particular group of women. Hence, this approach will lead you to a wrong answer. Had there been only 2 women in the above question, then your approach would have worked well. If we had 7 men and two women and the rest of the question is same, we would get an answer as:

Approach 1: 6! * 7C2 * 2! = 7! *6

Approach 2: 8! - 7!2! = 6*7!

I hope you get my point.

If you try to apply this approach, you have to first fix those two women who are together, which can be done in 5C2 ways. But when you are arranging them, there would be a case when three women W1W2W3 are together which is included multiple number of times. You have to subtract these cases, which is a very tedious process and hence unproductive.
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Re: Seven men and five women have to sit around a circular table  [#permalink]

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New post 18 May 2014, 08:21
mittalg wrote:
JusTLucK04 wrote:
Awesome approach Bunuel...
Why is this wrong neways: 12 Guys..No of ways 11!
Take 2 Women and bind them together...Now 11 guys and no of arrangements is 10!*2!
This covers all arrangements with at least 2 women together....

No of arrangements for no women together is: 11!- 10!*2! ---> wrong answer
Why?



The error is in 10!*2!. This is not taking into account which 2 women are together. This is always done when you know that which two persons are together. For example, if a question says that A & B should always be together, then only you can do it. Now they are not specifying any particular group of women. Hence, this approach will lead you to a wrong answer. Had there been only 2 women in the above question, then your approach would have worked well. If we had 7 men and two women and the rest of the question is same, we would get an answer as:

Approach 1: 6! * 7C2 * 2! = 7! *6

Approach 2: 8! - 7!2! = 6*7!

I hope you get my point.

If you try to apply this approach, you have to first fix those two women who are together, which can be done in 5C2 ways. But when you are arranging them, there would be a case when three women W1W2W3 are together which is included multiple number of times. You have to subtract these cases, which is a very tedious process and hence unproductive.


Whoops...!!!!
What a big blunder... :oops:
Thanks
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Re: Arrangement In a Circle  [#permalink]

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New post 18 May 2014, 09:53
NGGMAT wrote:

I understood this. But what i am trying to ask is that the qs is not saying that 2 men cannot be together. can we have an arrangement like below?


Yes, of course.
As a matter of fact, there would be twokinds of arrangement based on sexes :
One in which 3 men are together as shown by you.

Another one in which 2 men together appear twice .One such example shown as attached:

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Re: Seven men and five women have to sit around a circular table  [#permalink]

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New post 18 May 2014, 10:38
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I visualize these arrangements as following:
Attachment:
Roundarrangement.jpg
Roundarrangement.jpg [ 110.03 KiB | Viewed 3375 times ]
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Re: Seven men and five women have to sit around a circular table  [#permalink]

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New post 02 Jun 2014, 01:30
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jakolik wrote:
Seven men and five women have to sit around a circular table so that no 2 women are together. In how many different ways can this be done?


Quote:
the qs only says that the Women should not sit next 2 each other... but men can right? so y are we assuming

M_M_M_M_M_M_M_ : 14 Places

we can also have:

M_M_M_M_M_MMM: 12 places
MMM_M_M_M_M_M: 12 places

in this case the answer will be:

6!*5! right??

Also, i am not understanding how Bunnel got 21!


Yes, men can sit together but women cannot.

Don't assume places to be empty chairs. Think of a big round table. Each person who comes and sits around the table, brings his/her own chair along. Say the 7 men come and sit around the round table. They will be able to do that in 6! ways. Now, there is space between each pair of men. How many distinct spaces are there? 7 because there are 7 men say M1, M2, M3 till M7. So now you have empty space to the right of M1 and right of M2 and right of M3 etc. The women can take any 5 of these 7 spaces. Note that 2 women cannot take the same space because two women cannot sit together.

Say, the 5 women took 5 spots each to the right of M1, M2, M3, M4 and M5. So now spaces to the right of M6 and M7 are vacant. This means M6, M7 and M1 are sitting together with no one in between them. This takes care of the cases you have pointed out. So when we select 5 of the 7 spaces, we take care of all cases.

In how many ways can 7 men sit around a circular table? In 6! ways.
In how many ways that women select 5 of the 7 distinct places and arrange themselves in those places? In 7C5 * 5! ways.

Total arrangements = 6!* 7C5 * 5!
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Re: Seven men and five women have to sit around a circular table  [#permalink]

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New post 23 Sep 2014, 12:37
I did (7-1)! for the men (I placed them first). Then we place the women in the 7 slots. so 7P5.
I suppose when we talk about two men together, and the limited 5 women, 5 slots you mentioned above, that is a possible scenario. But aren't we trying to find the greatest possible means of placing the women? In that case, perhaps it would be better to create as many slots as possible and then place the women.
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Re: Seven men and five women have to sit around a circular table  [#permalink]

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Re: Seven men and five women have to sit around a circular table   [#permalink] 04 Sep 2019, 11:48

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