Chapter 13.2, Problem 24E

To determine

To find: Whether the arrangement of objects is a linear or circular and the number of arrangements for each situation.

Answer to Problem 24E

The arrangement is circular and the different ways of arranging $5$ members around a rectangular table is $24$ .

Explanation of Solution

Given information:A family of $5$ seated around a rectangular table.

The number of circular permutations of $n$ things taken all at a same time as $\left(n-1\right)!$ .

Since the family members are arranged in a circular manner so the arrangement is a circular permutation and the number of arrangements is given by using the formula $\left(n-1\right)!$ .

Here, $n$ is $5$ .

Substitute $5$ for $n$ in $\left(n-1\right)!$ .

  $\begin{array}{c}\left(5-1\right)!=4!\\ =4\times 3\times 2\times 1\\ =24\end{array}$$=$$4!$

Therefore, the number of different ways of arranging $5$ members around a rectangular table is $24$ .