<p>Find the number of ways that 8 beads of different colors be strung as a necklace.</p>
Step-by-Step Solution
Key Concept: A necklace is a circular arrangement where rotations are identical and reflections (flipping) are also identical. Use the formula for circular permutations with both rotational and reflectional symmetry: (n-1)!/2 for n distinct objects.
<p><strong>Step 1:</strong> Recognize that a necklace is a circular arrangement where we must account for rotational symmetry. For circular arrangements of n distinct objects, we fix one position to eliminate rotational duplicates, giving (n-1)! arrangements.</p><p><strong>Step 2:</strong> A necklace can be flipped over (reflected), which creates another symmetry. Two arrangements related by flipping are considered identical. Therefore, divide by 2 to account for reflectional symmetry.</p><p><strong>Step 3:</strong> For 8 beads of different colors: Number of arrangements = (8-1)!/2 = 7!/2 = 5040/2 = 2520</p><p>∴ Answer: <strong>2520</strong></p>
Correct Answer: 2520