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I hope someone can help me answer the following question. Thanks!

Here is a pseudo code of Permute-By-Sorting algorithm:

Permute-By-Sorting (A)

    n = A.length

    let P[1..n] be a new array

    for i = 1 to n

    P[i] = Random (1,n^3)

         sort A, using P as sort keys

In the above algorithm, the array P represents the priorities of the elements in array A. Line 4 chooses a random number between 1 and n^3.

The question is what is the probability that all priorities in P are unique? and how do I get the probability?

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By "all priorities are unique" I think you mean "all priorities are different." "Unique" does not mean "different." – bof May 15 at 23:31

You generate $n$ numbers in a range of $n^3$ numbers. There are

$$ \frac{n^3!}{(n^3-n)!} $$

favourable outcomes and a total of

$$ \left(n^3\right)^n $$

outcomes, so the probability is

$$ \frac{n^3!}{(n^3-n)!\left(n^3\right)^n}\;. $$

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