In how many ways can 6 beads be strung into a necklace ?

Options

A. a. 60
B. b. 360
C. c. 720
D. d. 120

Answer

To determine the number of ways to string 6 beads into a necklace, we need to consider the properties of circular arrangements and the fact that a necklace can be flipped.

### Step-by-Step Solution:

1. **Understanding Circular Arrangements**: When arranging items in a circle, we typically fix one item to eliminate the effect of rotations. For `n` items arranged in a circle, the number of arrangements is given by \((n - 1)!\).

2. **Applying the Formula**: For our case, we have 6 beads. Therefore, the number of arrangements in a circle is:
\[ (6 - 1)! = 5! \]

3. **Calculating Factorial**: Now, we calculate \(5!\):
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

4. **Considering Flipping**: Since a necklace can be flipped (i.e., it can be viewed from both sides), we need to divide the total arrangements by 2 to account for this symmetry:
\[ \text{Total arrangements} = \frac{5!}{2} = \frac{120}{2} = 60 \]

5. **Final Answer**: Thus, the total number of ways to string 6 beads into a necklace is:
\[ \boxed{60} \]