What is the probabiliy that a \(k\)-string over a set of size \(n\) forms a \(k\)-permutation? How does this question relate to the birthday paradox?

This is the probability that each element of our \(k\)-string is unique, and so this question is complementary to the birthday paradox:

\[\begin{split} Pr &= 1 \cdot \frac{n-1}{n} \cdot \frac{n-2}{n} \cdot \cdots \cdot \frac{n-k+1}{n} \\ &= \frac{(n-1)!}{(n-k)!n^k} \\ \end{split}\]