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Omni-MATH / We randomly choose a function f:[n]→[n], out of the n^(n) possible functions. We also choose an integer a uniformly at random from [n]. Find the probability that there exist positi…

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problem

We randomly choose a function f:[n]→[n]f:[n] \rightarrow[n], out of the nnn^{n} possible functions. We also choose an integer aa uniformly at random from [n][n]. Find the probability that there exist positive integers b,c≥1b, c \geq 1 such that fb(1)=af^{b}(1)=a and fc(a)=1f^{c}(a)=1. (fk(x)\left(f^{k}(x)\right. denotes the result of applying ff to xkx k times).
Plain-text mathematical notation (without MathML)
We randomly choose a function f:[n]→[n], out of the n^(n) possible functions. We also choose an integer a uniformly at random from [n]. Find the probability that there exist positive integers b,c≥1 such that f^(b)(1)=a and f^(c)(a)=1. (f^(k)(x) denotes the result of applying f to xk times).
Original LaTeX notation
We randomly choose a function $f:[n] \rightarrow[n]$, out of the $n^{n}$ possible functions. We also choose an integer $a$ uniformly at random from $[n]$. Find the probability that there exist positive integers $b, c \geq 1$ such that $f^{b}(1)=a$ and $f^{c}(a)=1$. $\left(f^{k}(x)\right.$ denotes the result of applying $f$ to $x k$ times).

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Source and history

task_id
81d8d547-d3c8-525f-a604-ca419e5d2002
task_key
test--81d8d547-d3c8-525f-a604-ca419e5d2002
task_revision_id
1
config / split
/ test

Official source

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