Question

In how many ways 4 married couples can be seated round a table if no husband and wife as well as two no two men are to be in adjacent seats.
A. 384
B. 14
C. 24
D. 36

Answer 1

Hint: To solve this problem we have to know about the concept of permutations and combinations. But here a simple concept is used. In any given word, the number of ways we can arrange the word by jumbling the letters is the number of letters present in the word factorial. Here factorial of any number is the product of that number and all the numbers less than that number till 1.
$ \Rightarrow n! = n(n - 1)(n - 2).......1$

Complete step-by-step solution:
Given there are four pairs of married couples, in which each couple contains one male and one female.
In a married couple, the male is called the husband and the female is the wife.
We have to arrange them in such a way that no husband and wife of the same marriage couple can sit beside each other and also men should not be seated beside each other. That means that no two men can be seated beside each other.
Here there are 4 men and 4 women, which makes 4 married couples.
Consider the 4 married couples be M1F1, M2F2, M3F3, M4F4. Where M stands for male and F stands for female.
Now no two men can be seated beside each other but these 4 men can be arranged in $4!$ways.
That is placing these 4 men alternatively, which can be arranged in $4!$ ways = $24$ ways.
$\because 4! = 4 \times 3 \times 2 \times 1$
$ \Rightarrow 4! = 24$
Let’s say we placed the 4 men like this:
M1_M2_M3_M4_.
Now as no couple can sit together so each blank has a choice of only 2 women.
Hence for each blank only 2 choices can be placed, as there are 4 blanks, the no. of ways in which the women are arranged as: $2 \times 2 \times 2 \times 2$
Thus the no. of ways 4 married couples can be seated round a table, where no husband and wife as well as two no two men are to be in adjacent seats, is given by:
\[ \Rightarrow 24 \times 2 \times 2 \times 2 \times 2\]
\[ \Rightarrow 24 \times 4 \times 4\]
\[ \Rightarrow 24 \times 16\]
\[ \Rightarrow 384\]
$\therefore $The no. of ways in which the 4 married couples can be seated round a table where no husband and wife as well as two no two men are seated beside is 384 ways.
The arrangement can be done in 384 ways.

Option A is the correct answer.

Note: Here while solving this problem one thing we have to understand is that why each blank is filled with only 2 women, for example consider the first blank, in that F1 or F2 can’t be placed as M1 is on the left of the blank which makes M1F1 a couple, and M2 is on the right of the blank which makes F2M2 a couple. Hence we can only place either F3 or F4, hence only 2 ways.