Question

Three married couples are to be seated in a row having six seats in a cinema hall. If spouses are to be seated next to each other, in how many ways they can be seated? Find also the number of ways of their seating if all ladies sit together.

Answer 1

Hint: Here, first we have to take the couples as A, B and C. First we have to calculate the number of possible ways that the couples are seated next to each other. Here, each couple has to be arranged among themselves in 2! ways. Each couple can be seated next to each other in 3! ways. With this data we have to calculate the total possible seating arrangements. Next, to find the number of ways that all ladies sit together, we have to arrange the ladies among themselves which is 3! ways and then, consider 3 men and 3 ladies as 4 units where 3 ladies are tied together which will be 4! Ways.

Complete Step-by-Step solution:
Here, we are given that the married couples are to be seated next to each other in a cinema hall.
Now, we have to find the number of possible ways they can be seated and also the number of ways of their seating if all ladies sit together.
First let us consider the three couples as A, B and C which each couple is taken as single unit as in figure:

Then the number of possible ways in which the spouses can be seated next to each other is 3! ways.
We also know that the couples can interchange their position and sit. Then the number of possible ways that the couple can be seated is 2! ways.
Since, there are three couples then each couple can be seated in 2! ways.
Therefore, the total number of possible ways that the couples sit next to each other is, n where:$n=3!\times 2!\times 2!\times 2!$
We know that:
$\begin{array}{rl}& 2!=1\times 2=2\\ & 3!=1\times 2\times 3=6\end{array}$
Now, by substituting these values we obtain:
$\begin{array}{rl}& n=6\times 2\times 2\times 2\\ & n=48\end{array}$
Therefore, we can say that the spouses can be seated next to each other in 48 possible ways.
Next, we have to find the number of ways of seating if all ladies sit together.
So, here we will have three ladies sitting together.
The three ladies among themselves are arranged in 3! ways.
Hence, we will have 4 units, 3 men and 3 ladies tied together. These 4 units can be arranged in 4! ways.
Therefore, the total number of possible ways where the ladies sit together = $4!\times 3!$
We have:
$\begin{array}{rl}& 4!=1\times 2\times 3\times 4=24\\ & 3!=1\times 2\times 3=6\end{array}$
Hence, the total number of possible ways where the ladies sit together = $24\times 6=144$
Therefore, we can say that there are 144 possible seating arrangements if all the ladies sit together.

Note: Here, we have to take the couples as a single unit. While arranging we also have to arrange among themselves which will be 2! ways. While arranging the ladies sitting together we will have 4 units, 3 men and 3 ladies tied together.