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How many distinct ways can 6 different keys be arranged on a circular key ring if flipping the ring is considered the same arrangement?

Mathematics

ScanSolve AI Answer

Answer

60

Step-by-step solution

5 steps

  1. Identify the number of distinct items.

    There are 6 different keys, so n=6n=6.

  2. Determine the type of arrangement.

    The keys are kept on a circular key ring. This is a circular permutation.

  3. Account for symmetry due to flipping.

    For arrangements on a circular key ring (or necklace), arrangements that are mirror images of each other are considered the same because the ring can be flipped over.

  4. Apply the formula for circular permutations with flipping.

    The number of distinct arrangements for nn distinct items on a circular key ring is given by the formula:

    (n−1)!2\frac{(n-1)!}{2}

  5. Substitute the value of nn and calculate.

    Number of arrangements=(6−1)!2\text{Number of arrangements} = \frac{(6-1)!}{2} Number of arrangements=5!2\text{Number of arrangements} = \frac{5!}{2} Number of arrangements=5×4×3×2×12\text{Number of arrangements} = \frac{5 \times 4 \times 3 \times 2 \times 1}{2} Number of arrangements=1202\text{Number of arrangements} = \frac{120}{2} Number of arrangements=60\text{Number of arrangements} = 60

    The final answer is 60.

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