Chapter 9.1, Problem 54E

Solution

To calculate: The numbers of seating arrangements that is possible for four persons sitting around a round table.

The resultant answer is 6 ways.

Given information:

There are four persons sitting around a round table.

Formula used: The number of permutations of n objects taken r at a time denoted by $P_{n_r}$ and given by $P_{n_r}=\frac{n!}{\left(n-r\right)!}$ .

Calculation:

Total number of persons is 4. Different numbers of seating arrangements are to be made for these four persons.

The expression can be written as,

  $P_{4_4}$

Rewrite the expression using the formula $P_{n_r}=\frac{n!}{\left(n-r\right)!}$ ,

  $\begin{array}{c}{P}_{{12}_6}=\frac{4!}{\left(4-4\right)!}\\ =\frac{4!}{0!}\end{array}$

Now expand the factorial,

  $\frac{4\cdot 3\cdot 2\cdot 1}{1}=24$

Since this is a round-table, the arrangements $ABCD,CDAB,\text{ and }DABC$ are the same for every arrangement.

Then number of ways of arranging 4 persons on a round table is: $\frac{24}{4}=6$

Therefore, the value of $P_{4_4}$ is 6. It means there are 6 ways in which the seating arrangements can be made for 4 persons.