Problem Solving

Seven women and seven men are to sit around a circular table such that there is a man on either side of every woman. What is the total number of seating arrangements?
(A) (7!)^2
(B) 6! X 7!
(C) (6!)^2
(D) 7!
(E) 6!

The women can be seated around a circular table in (7-1)!=6! ways
Now,there are seven vacant places in between all women where seven men can be seated in 7! ways.
[spoiler]Hence total no. of ways=6!7!
Option B[/spoiler]

firdaus117 wrote:The women can be seated around a circular table in (7-1)!=6! ways
Now,there are seven vacant places in between all women where seven men can be seated in 7! ways.
[spoiler]Hence total no. of ways=6!7!
Option B[/spoiler]

Hmm.. why would the women be seated around in 6! and not 7! ways? Shouldnt there be 7 options for the first woman, 6 for the next and so on?

arzanr wrote:

firdaus117 wrote:The women can be seated around a circular table in (7-1)!=6! ways
Now,there are seven vacant places in between all women where seven men can be seated in 7! ways.
[spoiler]Hence total no. of ways=6!7!
Option B[/spoiler]

Hmm.. why would the women be seated around in 6! and not 7! ways? Shouldnt there be 7 options for the first woman, 6 for the next and so on?

Circular arrangements(like pearls in a neclace or people around a circular table) are different than the straight line arrangements(arrangement of words or people in a row) in a way that in a circle there is no concept of a reference point or no concept of left-right.So we need a reference point around which we can arrange the things around.This is done by using one person as a reference.For example,we first sit a woman in the table then we build the arrangement by sitting other women aroung this particular lady.Hence since this lady is "fixed" as reference,other six can be moved around in 6! ways.
Hope I am clear.

A better explanation than above one:
Let's consider that 4 persons A,B,C, and D are sitting around a round table

Shifting A, B, C, D, one position in anticlock-wise direction, we get the following agreements:-




Thus, we use that if 4 persons are sitting at a round table, then they can be shifted four times, but these four arrangements will be the same, because the sequence of A, B, C, D, is same. But if A, B, C, D, are sitting in a row, and they are shifted, then the four linear-arrangement will be different.


Hence if we have '4' things, then for each circular-arrangement number of linear-arrangements =4

Similarly, if we have 'n' things, then for each circular - agreement, number of linear - arrangement = n.

Let the total circular arrangement = p

Total number of linear-arrangements = n.p

Total number of linear-arrangements

= n. (number of circular-arrangements)

Or Number of circular-arrangements = 1 (number of linear arrangements)

n = 1( n!)/n

circular permutation = (n-1)!