In how many ways four men, two women and one child can sit at a circul

Question

In how many ways four men, two women and one child can sit at a circular 7-seats table if the child is the only person to be seated between the two women?

(A) 24
(B) 36
(C) 48
(D) 96
(E) 240

Source: https://www.GMATH.net

HardikL · Answer

Let women be numbered as w1 and w2 and child be as c
Arrangement can be done as w1cw2 or w2cw1 I.e. 2 ways

Now group the women children as one so in addition to 4 other men there are 5 entities to be arranged in a circle which can be done in (n-1)! Ways = (5-1)!= 4!= 24 ways

So total ways = 2*24 = 48 ways

Posted from my mobile device

fskilnik · Answer

?\,\,\,:\,\,\,\,\# \,\,{	ext{circular}}\,\,{	ext{permutations}}\,\,{	ext{with}}\,\,{	ext{restrictions}}

Let the child be placed in any seat. 
Once this is done, there are 2 ways of placing the women (W1 to-the-right of the child, W2 to-the-left of the child... and vice-versa).
Once the child and the women are seated, there are 4! ways of placing the men.

From the  Multiplicative Principle :

{	ext{?}}\,\,\, = \,\,\,2 \cdot 4!\,\,\, = \,\,48

This solution follows the notations and rationale taught in the GMATH method.

Regards, 
Fabio.

fskilnik · Answer

?\,\,\,:\,\,\,\,\# \,\,{	ext{circular}}\,\,{	ext{permutations}}\,\,{	ext{with}}\,\,{	ext{restrictions}}

Alternate solution:

Let´s imagine a  linear version  (=row), but "connecting the first seat to the last one" (so that after the last seat we have again the first one).

https://preview.ibb.co/e0xyZ9/24_Set18_3z.gif

There are 7 seats in which the child could be seated.

Once (any) one of the 7 seats is chosen, there are 2 ways to seat the two women. 
(If the child is in the 7th seat, W1 will be in the 6th, W2 in the 1st... or vice-versa!)

Once the child and the two women are seated, there are 4! ways of seating the men.

Using the  Multiplicative Principle , we have 7*2*4! ways of seating these people  in the linear version .

The " linear to circular migration " is done dividing 7*2*4! by the number of objects to be circularized (7), 
checking the "connection" created earlier do not give rise to unwanted configurations: it does not! (*)

Hence:

? = \frac{{7 \cdot 2 \cdot 4!}}{7} = 48

(*) Typical problem: when A and B cannot stay next to each other, in the linear version you cannot allow one of them to be in 
the first place and the other in the last place, because when the connection is established they would violate the restriction!

This solution follows the notations and rationale taught in the GMATH method.

Regards, 
Fabio.

dabaobao · Answer

Official Solution  
Credit: Veritas Prep

We have 7 people and 7 seats around a circular table.

First let’s make the child sit anywhere in one way since all the places are identical. The two women can sit around the child in 2! ways. Now we have 4 distinct seats (relative to the people sitting) left for the 4 men and they can occupy the seats in 4! ways.

Total number of arrangements =1∗2!∗4!=48

Sneha333 · Answer

But in this question why we are not considering 2 women sitting together.This is also one case.

fskilnik · Answer

This question was asked by an online student of mine in 08 Aug 11 (see image attached) without mentioning the source.

I am happy to give Veritas the (previously unknown) credit for the excellent question.

Regards,
Fabio.

fskilnik · Answer

Hi  Sneha333 ,

This is not allowed in the question stem: "...the child is the only person to be seated between the two women".

Regards,
Fabio.

Sneha333 · Answer

Thanks for the reply Fabio.
My confusion is that as per the English in the question:If child is the only person that sits between 2 women.
So here there are 2 cases:1:If any one is between 2 women -->then the only possibility is child.
Case 2 :women are sitting together.

Do let me know your thoughts.

fskilnik · Answer

Hello again,  Sneha333 .

English is not my native language, but I believe that the phrase: "...the child is  the only  person  to be seated  between the two women." offers two important information/restrictions:

1.  There must be a person between the two women (and this person must be a child). 
2.  There is not a second person to be seated between the two women (because the person mentioned in 1. is the only one to satisfy this condition).

Regards,
Fabio.

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In how many ways four men, two women and one child can sit at a circul

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