Chapter 13.2, Problem 28E

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 28E

The arrangement is circular and the different ways of arranging the $9$ drinks along the edge of circular trey is $40,320$ .

Explanation of Solution

Given information: A waiter placing $9$ drinks along the edge of a circular trey.

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

Since the trey is in the form of circular hence the arrangement is circular permutation and the number of different ways of arranging $9$ drinks along the edge of a circular trey is given by using the formula $\left(n-1\right)!$ .

Here, $n=9$ .

Substitute $9$ for $n$ in $\left(n-1\right)!$ .

  $\begin{array}{c}\left(9-1\right)!=8!\\ =8\times 7\times 6\times 5\times 4\times 3\times 2\times 1\\ =40,320\end{array}$

Therefore, the number of different ways of arranging the $9$ departments along the outside perimeter of a supermarket is $40,320$ .