This classic probability puzzle takes on a new face:

You have 5 firms and 5 personalized cover letters, one for each.
The letters are randomly placed into the envelopes (each permutation equally likely).

Question: What is the probability that no letter ends up in the correct envelope?

This is a textbook case of the derangement problem—also known as the hat-check puzzle.

Step 1: Total Permutations

There are:

\[ 5! = 120 \text{ total permutations} \]

Each represents a way to stuff the 5 letters into 5 envelopes.

Step 2: Count Derangements

A derangement is a permutation with no fixed points (i.e., no item is in its correct position).

The number of derangements of \(n\) items is denoted by \( !n \), and for \(n = 5\):

\[ !5 = 44 \]

This can be computed by:

\[ !n = n! \left(1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \cdots + (-1)^n \frac{1}{n!} \right) \]

For \(n = 5\):

\[ !5 = 120 \left(1 - 1 + \frac{1}{2} - \frac{1}{6} + \frac{1}{24} - \frac{1}{120} \right) = 120 \times \frac{44}{120} = 44 \]

Step 3: Final Probability

\[ P(\text{no letter in correct envelope}) = \frac{!5}{5!} = \frac{44}{120} = \boxed{\frac{11}{30}} \]

Final Answer

The probability that none of the 5 letters ends up in the correct envelope is:

\[ \boxed{\frac{11}{30}} \]

Reference