Question 54
2024 girls each have their own doll. Consider all the ways of distributing the 2024 dolls to these girls so that each girl carries exactly one doll. For every $k \ge 0$, let $p(k)$ be the number of such ways in which exactly $k$ girls carry their own doll. Prove that
$$
\sum_{k=0}^{2024} k \times p(k) = 2024!.
$$
$$
\sum_{k=0}^{2024} k \times p(k) = 2024!.
$$