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4 couples are seating at a round tables how many ways can the 4 women

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keiraria
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4 couples are seating at a round tables how many ways can the 4 women [#permalink] New post   20 Apr 2012, 17:07
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4 couples are seating at a round tables. In how many ways can the 4 women sit together?

A. 4!
B. 5!
C. 4!*4!
D. 5!*4!
E. 5!*5!
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Bunuel
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Re: 4 couples are seating at a round tables how many ways can the 4 women [#permalink] New post   12 Jul 2012, 01:39
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manasrastogi wrote:
Can we put all the women in 1 group.
Total will be : 4 men + 1 group of women = 5
To arrange these 5, we will have 5! ways.
Again to group all the women we will have = 4! ways.
Then total ways = 5! X 4!


The red part is not correct.

Notice that we have circular not linear arrangement.

The difference between placement in a row and that in a circle is following: if we shift all object by one position, we will get different arrangement in a row but the same relative arrangement in a circle. So, for the number of circular arrangements of n objects we have: \(\frac{n!}{n}=(n-1)!\).

Hence 4 men and 1 group of four women (total of 5 units) can be arranged around a circular table in (5-1)!=4! ways. Next, these 4 women within their unit can be arranged in 4! ways, which gives total equal to 4!*4!.

Questions about this concept to practice:
in-how-many-different-ways-can-4-ladies-and-4-gentlemen-be-102187.html
at-a-party-5-people-are-to-be-seated-around-a-circular-104101.html
seven-family-members-are-seated-around-their-circular-dinner-102184.html
seven-men-and-seven-women-have-to-sit-around-a-circular-11473.html
a-group-of-8-friends-sit-together-in-a-circle-alice-betty-106928.html
seven-men-and-five-women-have-to-sit-around-a-circular-table-98185.html
a-group-of-8-friends-sit-together-in-a-circle-alice-betty-106928.html
find-the-number-of-ways-in-which-four-men-two-women-and-a-106919.html

Hope it helps.
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Bunuel
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Re: 4 couples are seating at a round tables how many ways can the 4 women [#permalink] New post   21 Apr 2012, 00:00
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keiraria wrote:
hello

4 couples are seating at a round tables
how many ways can the 4 women sit together

please explain your reasoning


best regards

source : saskschools.ca
I am trying to understand circular permutation


4 men can be arranged around a table in (4-1)!=3! ways;

Now, we can place 4 women in 4 slots between any two men and arrange these women in this slot in 4! ways.

So, total # of ways is 3!*4*4!=4!*4!

Another way: if 4 women sit together around a table then 4 men also sit together, so total # of arrangements is 4!*4!.

Check this for circular arrangements: math-combinatorics-87345.html

Hope it helps.
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Re: 4 couples are seating at a round tables how many ways can the 4 women [#permalink] New post   11 Jul 2012, 22:46
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Can we put all the women in 1 group.
Total will be : 4 men + 1 group of women = 5
To arrange these 5, we will have 5! ways.
Again to group all the women we will have = 4! ways.
Then total ways = 5! X 4!
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manasrastogi
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Re: 4 couples are seating at a round tables how many ways can the 4 women [#permalink] New post   12 Jul 2012, 02:20
Thanks Bunuel.....
I got it and thanks for posting additional links for practice.
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KarishmaB
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4 couples are seating at a round tables how many ways can the 4 women [#permalink] New post   Updated on: 08 Oct 2022, 22:15
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Check out posts on this concept on the blog link given in my signature below.

Originally posted by KarishmaB on 12 Jul 2012, 21:47.
Last edited by KarishmaB on 08 Oct 2022, 22:15, edited 1 time in total.
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Re: 4 couples are seating at a round tables how many ways can the 4 women [#permalink] New post   07 Jan 2024, 22:13
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Re: 4 couples are seating at a round tables how many ways can the 4 women [#permalink]
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