Discrete Mathematics Level 3

At a recent magic show, the magician has 10 cards, each assigned a distinct integer from 1 to 10 (inclusive). After randomly shuffling the cards, he claims that they are now in order, which the audience disbelieves. Let \(p\) be the probability that exactly 9 of the cards are arranged in order. What is the value of \( \lfloor 1000p \rfloor \)?

Details and assumptions

The function \(\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z}\) refers to the greatest integer smaller than or equal to \(x\). For example \(\lfloor 2.3 \rfloor = 2\) and \(\lfloor -5 \rfloor = -5\).

The card is in order if it's position in the sequence corresponds to the numerical value of the card. As an explicit example, the sequence 1 , 3, 5, 7, 9, 10, 6, 8, 2, 4 has 2 cards (listed in bold) that are in order.


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