Question:

This is a derangement problem where we need to count permutations of 7 distinct hats among 7 persons such that no person gets their own hat. The number of derangements $D_n$ is given by $D_n = n! \sum_{i=0}^{n} \frac{(-1)^i}{i!}$. For $n=7$: $D_7 = 7! \left(1 - 1 + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!} + \frac{1}{6!} - \frac{1}{7!}\right) = 5040 \times 0.3679 = 1854$. Alternatively, using the recurrence $D_n = (n-1)(D_{n-1} + D_{n-2})$ with $D_1=0, D_2=1$ gives the same result.