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What this combinatorial counting puzzle tests

This is a hard combinatorial puzzle that requires careful constraint management. It tests your ability to break a multi-part constraint into mutually exclusive cases and count each rigorously, without double-counting or missing valid configurations.

To solve problems of this type, you need to identify which structures satisfy all the conditions simultaneously: each person must make a choice, a global property must hold (exactly one course has overlap), and you must account for the symmetries in the problem. The difficulty lies not in any single calculation but in ensuring your case breakdown is exhaustive and non-overlapping, then verifying that each case satisfies all stated constraints.

• Partitioning by constraint structure
• Multinomial and Stirling number reasoning
• Checking for hidden symmetries and symmetry-breaking
• Verification that all cases are consistent with the problem statement