In how many ways can 30 different pearls be arranged to form a necklace?

Answer

To solve the problem of how many ways 30 different pearls can be arranged to form 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. **Understand the Problem**: We have 30 different pearls and we want to arrange them in a circular manner to form a necklace. Since a necklace can be flipped, arrangements that are mirror images of each other are considered the same.

2. **Use the Formula for Circular Permutations**: For arranging \( n \) distinct objects in a circle, the number of arrangements is given by \((n - 1)!\). This is because one object can be fixed to break the circular symmetry.

\[ \text{Circular arrangements} = (n - 1)! \]

3. **Account for Flipping**: Since a necklace can be flipped, we need to divide the number of arrangements by 2. Therefore, the formula for the number of arrangements of pearls in a necklace becomes:

\[ \text{Necklace arrangements} = \frac{(n - 1)!}{2} \]

4. **Substitute the Value of \( n \)**: Here, \( n = 30 \). So we substitute \( n \) into the formula:

\[ \text{Necklace arrangements} = \frac{(30 - 1)!}{2} = \frac{29!}{2} \]

5. **Final Calculation**: The final answer for the number of ways to arrange 30 different pearls to form a necklace is:

\[ \frac{29!}{2} \]

### Conclusion:
The number of ways to arrange 30 different pearls to form a necklace is \(\frac{29!}{2}\).

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