1. The problem is to generate 500 sets, each containing 12 unique numbers from 1 to 30, such that each number appears equally often with every other number, and no set is repeated. 2. This is a combinatorial design problem related to Balanced Incomplete Block Designs (BIBD), where each pair of elements appears together in the same number of sets. 3. The parameters are: $v=30$ (total numbers), $k=12$ (numbers per set), $b=500$ (number of sets), and we want each pair to appear together equally often. 4. The key BIBD equation is $$r(k-1) = \lambda (v-1)$$ where $r$ is the number of sets each element appears in, and $\lambda$ is the number of sets each pair appears in. 5. Also, $$bk = vr$$ since total elements counted across all sets equals total appearances. 6. We check if such a design exists with these parameters. From $bk=vr$, we get $$r = \frac{bk}{v} = \frac{500 \times 12}{30} = 200$$ meaning each number appears in 200 sets. 7. Using $r(k-1) = \lambda (v-1)$, substitute $r=200$, $k=12$, $v=30$: $$200 \times 11 = \lambda \times 29 \Rightarrow 2200 = 29\lambda \Rightarrow \lambda = \frac{2200}{29} \approx 75.86$$ 8. Since $\lambda$ is not an integer, a perfect BIBD with these parameters does not exist. 9. Therefore, it is impossible to generate 500 sets of 12 numbers from 30 such that each pair appears exactly the same number of times. 10. One can try approximate or heuristic methods, but exact equal pair appearances with these parameters is not achievable.