Chapter 9.6, Problem 27E

(a)

To determine

The total number different ways in which four couples can sit

Answer to Problem 27E

40320 ways

Explanation of Solution

Given:

There is no sitting restriction

Calculation:

Number of ways in which 4 couples can be arranged with no sitting restrictions will be ${}_8{P}_8$

  $\begin{array}{l}=\frac{8!}{(8-8)!}\\ =40320\end{array}$

(b)

To determine

The total number different ways in which four couples can sit

Answer to Problem 27E

384 ways

Explanation of Solution

Given:

The two members of each couple will sit together

Calculation:

Now considering two members of a couple as 1, there are 4 persons and 4 seats available.

Number of ways in which 4 couples can be arranged with no sitting restrictions will be ${}_4{P}_4$

Consider the couples in a specific arrangement when male sits to the right of the lady. Now other arrangements would be that either one of the couple sits with male to the left of the lady or two couple with male on the left and so on.

This leads to ${}_4{C}_0{\times }_4{C}_1{\times }_4{C}_2{\times }_4{C}_3{\times }_4{C}_4$ ways of arranging couples among themselves

Hence, from Fundamental Counting Principle,

  $\begin{array}{c}{}_4{P}_4\times \left({}_4{C}_0{\times }_4{C}_1{\times }_4{C}_2{\times }_4{C}_3{\times }_4{C}_4\right)=\frac{4!}{0!}\times \left({\displaystyle \sum _{i=0}^4{}_4{C}_i}\right)\\ =24\times \left({\displaystyle \sum _{i=0}^4\frac{4!}{\left(4-i\right)!i!}}\right)\\ =24\times \left(1+4+6+4+1\right)\\ =24\times \left(16\right)\\ =384\end{array}$