Question

Four men, two women and a child sit at a round table. Find the number of ways of arranging the seven people if the child is seated
(a) between these two women.
(b) between two men.

Answer 1

Answer:

a) 48 ways

b) 288 ways.

Step-by-step explanation:

According to the question there are 4 men and 2 women and a child.

In the first case you have to find out the number of ways in which the child is to be seated between the two women. It means that we have to always place two women and a child in between them. It will have two ways in case the women exchanges their seat. And the remaining four men can sit anywhere that is 4 factorial.

All together we have 2 * 4! = 2 * 24 = 48 ways.

Now, when we have to find the number of way in which the child can be seated between two men. We will proceed in the very similar way we will place a child in between two men and that two men will be selected randomly and the number of arrangement that is permentation will be

$\frac{4!}{2!}$ = 24 / 2 = 12 ways.

Now the remaining two men and 2 women can be again arranged in 4! ways.

So, altogether we have, 4! * 12 = 288 ways.

Answer 2

Answer:

a) 48

b)288

Step-by-step explanation:

Hi,

Given 4 men , 2 women and a child who has to be seated around a round

table.

In general n persons can be seated around a round table in (n-1)! ways

a) If child has to be seated between 2 women, let us consider all 3 as

together, then we need to arrange 4 men and this group at round table,

this can be done in 4! ways = 24 ways.

But 2 women can  interchange their positions among themselves which can

be done in 2 ways, hence total number of ways would be 24 *2 = 48 ways

b) If child has to be seated between 2 men, let us choose any 2 men in

between whom child would be seated, so selected of 2 men could be done

in 4C2 ways = 6 ways, now all these 3 as group we can place 2 men + 2

women + group of 3 at round table in 4! ways = 24 ways, but these 2 men

can be interchange their positions among themselves which can be done in

2 ways, hence total number of ways would be 6*24 *2 = 288 ways

Hope, it helps !