Question

Three married couples have purchased six seats in a row for musical comedy. Number of different ways which can they be seated, if each couple must sit together, is
A. $48$
B. $6$
C.$8$
D. $9$
E.$12$

Answer 1

Hint: First, we learn the meaning of the terms permutation and combination which are important topics in probability.
In Probability, the permutation is the process of arranging the outcomes in order. Here, the order must be followed to arrange the items.
In Probability, the term combination refers to the process of selecting the outcomes in which the order does not matter. Here, the order to arrange the items is not followed.

Complete step by step answer:
 It is given that there are three married couples and they have purchased six seats in a row for musical comedy.
Here, we need to calculate the number of different ways in which they can be seated, if each couple must sit together.
It can be clearly understood that there are three couples and three pairs of seats.
Let us deal with it briefly.
The first couple has choices to choose their seats from three pairs. The remaining two options will be chosen by the second couple and the last will be chosen by the third couple.
Hence, the total choices for the couples are$3! = 3 \times 2 \times 1$$ = 6$ ways…. $\left( 1 \right)$
Here, each couple can swap their seats without sitting apart. Also, three couples can swap their seats. The number of ways in which each husband and wife can swap their seats is $2! \times 2! \times 2! = 8ways$…. $\left( 2 \right)$
Hence, by multiplying the above equations, we can get the required total number of arrangements.
The required total number of arrangements, $ = 6 \times 8$ $ = 48ways$

So, the correct answer is “Option A”.

Note: The formula to find the permutation is as follows.
\[{}_n{P_r} = n(n - 1)(n - 2).......(n - r + 1)\]
$ = \dfrac{{n!}}{{(n - r)!}}$ (! Is a mathematical symbol called the factorial)
Where $n$ denotes the number of objects from which the permutation is formed and $r$ denotes the number of objects used to form the permutation.
Now, the formula to calculate the combination is as follows.
${}_n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$
Where $n$ denotes the number of objects from which the combination is formed and $r$ denotes the number of objects used to form the combination.