Permutations of permutations

What this permutation-counting puzzle tests

This is a medium-difficulty combinatorics puzzle that rewards symmetry-based reasoning over brute enumeration. It appears in quant interviews to see whether candidates can spot structural properties that simplify counting, rather than grinding through cases.

The core skill being tested is the ability to recognize when two classes of objects are in bijection—that is, when a clever pairing or symmetry argument can reduce a hard count to an easier one. Rather than listing permutations by hand or writing out inclusion-exclusion, strong solutions identify an invariant or a one-to-one correspondence that makes the answer fall out immediately.

• Symmetry arguments in combinatorics
• Bijective proof techniques
• Complement and direct counting