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How many permutations of the numbers 1,2,...,9 have exactly two fixed…
Question
How many permutations of the numbers 1,2,...,9 have exactly two fixed points (a fixed point is a position i with the value i)?
✓ Verified answer: 66744checked by our engine — not a guess
Step-by-step solution
Choose which 2 of the 9 positions are the fixed points: C(9,2) = 36 ways.
The remaining 7 elements must be deranged (no further fixed points), which is the derangement number D(7).
Compute D(7) = 7! * (1 - 1/1! + 1/2! - 1/3! + 1/4! - 1/5! + 1/6! - 1/7!). Numerically D(7) = 1854.
Total = 36 * 1854 = 66744.
Final answer66744
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