Exercise 10. Count the number of rearrangements $a_{1}, a_{2}, \ldots, a_{2023}$ of the sequence $1,2, \ldots, 2023$ such that $a_{k}>k$ for exactly one value of $k$.
Solution to Exercise 10: To a valid rearrangement, we can associate a subset of $1, \ldots, 2023$ of cardinality at least 2: the set of $k$ such that $a_{k} \neq k$.

We can show that to a subset of cardinality at least 2 of $1, \ldots, 2023$, we can associate a unique valid rearrangement that corresponds to it. If the set is $E=\left\{e_{1}, \ldots, e_{n}\right\}$ with $e_{1}<e_{2}<\cdots<e_{n}$, the unique rearrangement that has $E$ as its associated set satisfies: $a_{e_{1}}=e_{n}, a_{e_{2}}=e_{1}, a_{e_{3}}=e_{2}, \ldots, a_{e_{n}}=e_{n-1}$ and for all $k \notin E, a_{k}=k$.

Thus, the number of valid rearrangements is equal to the number of subsets of $1, \ldots, 2023$ of cardinality at least 2. The answer is therefore:

$$
2^{2023}-2024
$$

Grader's Comment: Most of the submitted solutions are correct. Only two reasoned as in the solution. The others either reasoned by induction by replacing 2023 with $n$, or proceeded by summation (two submissions presented the correct sum but did not know how to compute it: perhaps they do not know the sum of a geometric series)