1. The problem is to create 500 sets, each containing 12 random numbers selected from a set of 30 numbers. 2. Each number must appear the same number of times across all sets. 3. Additionally, no set should be repeated. 4. Let's analyze the constraints mathematically: - Total numbers in all sets combined: $500 \times 12 = 6000$. - Since there are 30 unique numbers, and each must appear equally often, each number appears $\frac{6000}{30} = 200$ times. 5. The challenge is to arrange these 6000 numbers into 500 unique sets of 12 numbers each, with each number appearing exactly 200 times. 6. This is a combinatorial design problem related to balanced incomplete block designs (BIBD), where each element appears equally often and sets (blocks) are unique. 7. Constructing such a design manually or randomly is complex and typically requires algorithmic or combinatorial design methods. 8. A practical approach is to use algorithms that generate balanced combinations ensuring equal frequency and uniqueness, such as backtracking or heuristic search. Final answer: It is mathematically feasible since $30 \times 200 = 6000$ matches total elements needed, but requires combinatorial design or algorithmic generation to ensure equal frequency and no repeated sets.