In how many different ways can 20 different pearls be arranged to form a necklace?

Answer

To find the number of different ways to arrange 20 different pearls to form a necklace, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Arrangement**:
A necklace is circular, which means that arrangements that can be rotated into each other are considered the same.

2. **Fixing One Pearl**:
To simplify the calculation, we can fix one pearl in place. This means we will arrange the remaining 19 pearls around it.

3. **Calculating Arrangements**:
The number of ways to arrange 19 pearls in a circular manner is given by (n-1)!, where n is the number of pearls. Here, n = 20, so we have:
\[ \text{Arrangements} = (20 - 1)! = 19! \]

4. **Accounting for Symmetry**:
Since a necklace can be flipped (i.e., it can be viewed from both sides), each arrangement is counted twice (once for each direction). Therefore, we need to divide the total arrangements by 2:
\[ \text{Final Count} = \frac{19!}{2} \]

5. **Final Calculation**:
Now we can compute the value of \(19!\) and then divide by 2 to get the final answer.

### Final Answer:
\[ \text{Number of different ways} = \frac{19!}{2} \]