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Part BCSIR NET June 2024exactly-k-fixed-needs-derangement

Exactly k fixed needs derangement

How many arrangements of the digits of the number 1234567 are there, such that exactly three of them occur in their original position? (E.g., in the arrangement 5214763, exactly the digits 2, 4 and 6 are in their original positions. In the arrangement 1243576, exactly the digits 1, 2 and 5 are in their original positions.)

  1. A.525
  2. B.35
  3. C.840
  4. D.315✓

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

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50 are analysed free — try those first.

Related counterexample: Aₙ is simple for every n ≥ 3

More on this topic

The chapter behind this: Permutation groups — cycle type, sign and conjugacy — free to read

From Groups › Permutation groups: cycles, sign, conjugacy in S_n and A_n

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