Question

Let us consider m be the number of ways in which two couples can be seated on $4$ chairs in a row so that no wife is next to her husband and n be the number of ways in which they can be seated in $4$ chairs in a circle In the second case, rotations are considered for different configurations. So what will be the condition of the configuration . Choose the correct answer
(A) \[m = n\]
(B) \[m = 2n\]
(C)\[m = 5n\]
(D) \[m = 8n\]

Answer 1

Hint:Making an arrangement of seats we have two ways first situation is in row wise but the condition is we no wife sit next to her husband for this we have two situations. For the second situation of arrangement we made it circular for this we also have two situations. By comparing both the ways we got our answer.

Complete step-by-step answer:
According to the question we have to make a sitting arrangement for two couples in which no wife will sit next to her husband.
So there are two ways: one way is row wise and the second way is circular.
m be the number of ways in row wise arrangement and n be the number of ways in circular arrangement.

For first situation which is row wise, there are two choices
For this let H1 be the husband of the first couple, W1 be wife of first couple, H2 be the husband of second couple, W2 be wife of the second couple.
So we have two situations: first is H1 then W2 then W1 and H2.
Second situation is W1 then H2 then H1 and W2.
Therefor the value of the number of ways in row wise \[m = 2\]

For second situation which is circular, we have 2 choices
They first is in circular arrangement we go clockwise i.e. H1 then W2 then W1 and H2.
Second situation we go clockwise i.e. H2 then W1 then W2 and H1.
Therefor the value of the number of ways in circular is \[n = 2\]
Hence the value of m and n are equal i.e. \[m = n\]

So, the correct answer is “Option A”.

Note: For solving these types of questions we should remember to make an arrangement considering the type whether it is row wise or circular and also consider the condition which is given in the question and then we solve the question and make a diagram of the situations.