1. The problem is to create 500 sets of 12 random numbers each from the numbers 1 to 30, such that each number appears the same number of times 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 subsets. 3. The parameters are: total elements $v=30$, block size $k=12$, number of blocks $b=500$. 4. For a BIBD, the number of blocks $b$, the number of times each element appears $r$, and the number of times each pair appears $\\lambda$ satisfy: $$r(k-1) = \\lambda (v-1)$$ $$bk = vr$$ 5. We want to find integers $r$ and $\\lambda$ satisfying these equations with $b=500$, $v=30$, $k=12$. 6. From $bk=vr$, we get $500 \times 12 = 30r \Rightarrow 6000 = 30r \Rightarrow r=200$. 7. From $r(k-1) = \\lambda (v-1)$, 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 create 500 sets of 12 numbers from 30 elements where each pair appears exactly the same number of times. 10. Alternative approaches involve approximate or heuristic methods, but exact balanced designs are not possible with these parameters. Final answer: A perfect balanced design with 500 sets of 12 numbers from 30 elements where each pair appears equally is not possible.