HMMT 2025 U18

Let $f:\{1,2,3,\ldots ,9\} \to \{1,2,3,\ldots ,9\}$ be a permutation chosen uniformly at random from the 9! possible permutations. Compute the expected value of $\underbrace{f(f(\cdots f(f(1))\cdots))}_{2025 f s}$ .