The Number of Seniors and Juniors to Join the Company

Count how many ways to assign employees or positions as seniors or juniors subject to per-position, pairwise, or global constraints (e.g., minimum/maximum seniors, compatibility limits, or required totals). The challenge is efficient combinatorial counting under these constraints for large inputs, typically solved with dynamic programming, combinatorics (binomial coefficients/inclusion–exclusion) and modular arithmetic or flow-like reductions.