Chapter 9.1, Problem 55E

Solution

To state: The numbers of arrangements that is possible for four beads (red, blue, yellow, and green) that are to be arranged on a string to make a simple necklace.

The resultant answer is 6 ways.

Given information:

Four beads (red, blue, yellow, and green) are to be arranged on a string to make a simple necklace.

Explanation:

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

In a circle, there is no distinction between the last and first place, only changes in relative position is required.

Therefore, fix one of the four beads in a fix place and then arrange the remaining $\left(4-1\right)$ beads in $\left(4-1\right)!$ .

Then the value of $\left(4-1\right)!$ is:

  $\begin{array}{c}3!=3\cdot 2\cdot 1\\ =6\end{array}$

Therefore, there are 6 ways in which the 4 beads can be arranged in a string.