Mismatched Letters Problem (Derangement) - Lycée de la Salle (Alès)

An individual has written \( k \) letters to each of their \( k \) different friends, and addressed the corresponding \( k \) envelopes.
How many different ways are there to place every letter into an \textbf{incorrect} envelope?
In other words, how many permutations of \( k \) elements have \textbf{no fixed point}?