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Finding Unique Necklaces with Different Colored Beads - Combinatorics Problem
Published on September 27, 2026
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Calculate the number of unique necklaces that can be made using 7 differently colored beads by applying circular permutation and reflection symmetry formulas. The correct answer is 360.
Question
Answer
- First, calculate the number of linear permutations of 7 different colored beads, which is
:
- Since the beads are on a necklace, they are arranged in a circle. To account for rotational symmetry, we divide the number of linear permutations by the number of beads (
):
- Because a necklace can be flipped over (reflectional symmetry), we must also divide by 2 to find the number of unique necklaces:
The correct answer is D. 360.
Theorems
• Circular Permutation
The number of ways to arrange
• Necklace Permutation
When necklaces can be flipped over, the number of unique arrangements is halved due to reflectional symmetry, calculated as