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: Start with the circular count.
Arranging $n$ different items in a ring gives $(n-1)!$ ways. With $8$ pearls that is $7!$.

Step 2: Compute the factorial.
$7! = 5040$.

Step 3: Account for flipping.
A necklace can be turned over, so each arrangement and its mirror look the same. We divide by $2$.

Step 4: Final count.
$\tfrac{5040}{2} = 2520$. \[ \boxed{2520} \]