Question

The number of ways in which 6 gentlemen and 3 ladies are seated round a table so that every gentleman may have a lady by his side is?
Option A: 1440
Option B: 720
Option C: 240
Option D: 480

Answer 1

The most important formula when it comes to circular permutation is that:
When there are $n$ people, they have to be arranged around a circular table, the total number of ways of arranging them are: $\dfrac{{n!}}{n} = (n - 1)!$

Complete step-by-step answer:
We have to arrange 6 gentlemen and 3 ladies such that every gentleman has a lady by his side.
We can start by first arranging the men around the table.
The number of ways it can be done is: $(6 - 1)! = 5! = 120$
Now, we can place the three ladies in such a way that every man has a lady on either of his side.
The number of ways of arranging the ladies is: $3! = 6$
The number of ways of arranging both men and ladies is: $120 \times 6 = 720$
Since, their left and right sides can be interchanged but still the given condition will be obeyed, the total number of ways the given arrangement can be done is:
(Number of ways of arranging men) $\times$ (number of ways of arranging women)
$720 \times 2 = 1440$

So, the correct answer is “Option A”.

Note: When the things or people are arranged around a circle, then it’s called circular permutation. We have to remember the formula of circular permutation which is: $\dfrac{{n!}}{n} = (n - 1)!$ .
If the clockwise and anti-clockwise can be distinguished, for example when it involves people, we have to multiply it with 2.
If the clockwise and anti-clockwise can not be distinguished, for example when it involves flowers, multiplication with 2 is not required.