This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Identify the number of distinct items. There are 6 different keys, so n=6. Step 2: Determine the type of arrangement. The keys are kept on a circular key ring. This is a circular permutation. Step 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. Step 4: Apply the formula for circular permutations with flipping. The number of distinct arrangements for n distinct items on a circular key ring is given by the formula: ((n-1)!)/(2) Step 5: Substitute the value of n and calculate. Number of arrangements = ((6-1)!)/(2) Number of arrangements = (5!)/(2) Number of arrangements = (5 × 4 × 3 × 2 × 1)/(2) Number of arrangements = (120)/(2) Number of arrangements = 60 The final answer is 60.