Chapter 13.2, Problem 26E

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 26E

The arrangement is linear and the different ways of arranging the $6$ houses on a cul-de-sac to the incoming street is $720$ .

Explanation of Solution

Given information: $6$ houses on a cul-de-sac relative to the incoming street.

The number of linear permutations of $n$ things taken all at a same time as $n!$ .

Since the houses on a cul-de-sac relative to the incoming street hence the arrangement is linear permutation and the number of different ways of arranging the $6$ houses on a linear is $n!$ .

Here, $n=6$ .

Substitute $6$ for $n$ in the given formula $n!$ .

  $\begin{array}{c}6!=5!\\ =6\times 5\times 4\times 3\times 2\times 1\\ =720\end{array}$

Therefore, the number of different ways of arranging the $6$ houses on a cul-de-sac to the incoming street is $720$ .