1. Problem statement. 1. Generate 500 distinct subsets of size 12 from the set $\{1,2,\dots,30\}$ so that every pair of distinct numbers appears together the same number of times. 2. Modeling and key idea. 2. Model this as a balanced incomplete block design (BIBD) with parameters $v=30$, $b=500$, $k=12$, where each element appears in $r$ blocks and each pair appears in $\lambda$ blocks. 3. Fundamental BIBD formulas and rules. 3. The standard relations for a BIBD are $$vr= bk$$ and $$\lambda(v-1)= r(k-1)$$ These equations must hold with integer $r$ and integer $\lambda$ for an exact pairwise-balanced design. 4. Compute $r$ from $b$, $k$, $v$. 4. Use the formula $r=\frac{bk}{v}$ to compute $$r=\frac{500\cdot 12}{30}=200$$ So each element would need to appear in $r=200$ blocks. 5. Compute $\lambda$ and check integrality. 5. Use $\lambda=\frac{r(k-1)}{v-1}$ to compute $$\lambda=\frac{200\cdot 11}{29}=\frac{2200}{29}\approx 75.8621$$ Since $\lambda$ is not an integer, an exact BIBD with $v=30$, $k=12$, $b=500$ cannot exist. 6. Algebraic condition for integer $\lambda$ and implications. 6. Combine formulas to get $$\lambda= b\cdot \frac{k(k-1)}{v(v-1)}$$ For $v=30$ and $k=12$ we have $$\frac{k(k-1)}{v(v-1)}=\frac{12\cdot 11}{30\cdot 29}=\frac{132}{870}=\frac{22}{145}$$ Hence $$\lambda= b\cdot \frac{22}{145}$$ Because $\gcd(22,145)=1$, $b$ must be a multiple of 145 for $\lambda$ to be integer. 7. Smallest exact-balanced choices and comparison to 500. 7. The smallest positive $b$ giving integer $\lambda$ is $b=145$, which yields $$r=\frac{145\cdot 12}{30}=58$$ and $$\lambda=\frac{145\cdot 22}{145}=22$$ Multiples of 145 also give integer parameters, e.g., $b=290$, $b=435$, $b=580$, etc. Because 500 is not a multiple of 145, the exact requirement cannot be satisfied for $b=500$. 8. Practical alternatives to achieve near-uniform pair frequencies. 8. Option A (change the number of sets): choose $b$ equal to a multiple of 145 near 500, e.g., $b=435$ (less) or $b=580$ (more), and construct or search for a BIBD if it exists for those parameters. 8. Option B (approximate balanced solution): generate 500 distinct random 12-subsets and use iterative local search or simulated annealing to minimize the variance of pairwise co-occurrence counts. 8. A simple greedy heuristic is: start with an empty collection, repeatedly sample candidate 12-subsets not yet chosen, compute the resulting pair-count variance if the candidate is added, and pick candidates that minimize variance; continue until you have 500 unique sets. 8. Option C (optimization): formulate an integer program with binary variables $x_S$ for each 12-subset $S$, constrain $\sum_S x_S=500$ and minimize sum of squared deviations of pair counts from their mean; solve with a MIP solver or approximate solver. 9. Recommended takeaway and final answer. 9. Final answer: An exact design with the stated property is impossible for $b=500$ because $\lambda=\frac{2200}{29}$ is not an integer. 9. To get exact equal pair frequencies you must pick $b$ a multiple of 145, for example $b=145$ or $b=435$ or $b=580$; otherwise use heuristic or optimization methods to produce 500 distinct sets with near-uniform pair counts.