Sort algorithm input probabilities
Here is why you would expect these properties to hold. Suppose that $a_1,\ldots,a_n$ are chosen independently from the uniform distribution on $[0,1]$. The event $a_i = a_j$ has probability zero, and so with probability $1$ all numbers are distinct. Moreover, given that, all sequence orderings are equally likely, and since there are $n!$ of them, the probability that the sequence is ordered is exactly $1/n!$.

To prove that these claims hold in the limit even when the $a_i$ are sampled from a finite set requires some calculation, which I leave to you.