Question

The number of ways in which 8 different pearls can be arranged to form a necklace is

• A. 40320
• B. 5040
• C. 2520
• D. 1260

Hint:

Always distinguish between circular arrangements of people (where left and right are fixed) and circular arrangements of objects like beads or pearls (where flipping the necklace doesn't change the relative order).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to arrange 8 distinct items in a circular arrangement specifically designed to be a necklace.

Step 2: Key Formula or Approach:
For $n$ distinct items, the number of circular arrangements is $(n-1)!$. For a necklace, since the necklace can be flipped (clockwise and anti-clockwise arrangements are considered identical), the formula is $\frac{(n-1)!}{2}$.

Step 3: Detailed Explanation:
Here, $n = 8$.
First, calculate the circular permutations: $(8 - 1)! = 7!$.
$7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040$.
Since it is a necklace, we divide by 2 to account for reflectional symmetry:
$\frac{5040}{2} = 2520$.

Step 4: Final Answer:
The number of ways to arrange the pearls is 2520, which corresponds to option (C).