easy Probability

The Fixed One I

A release-checked easy problem for training Combinations and Permutations, Conditional Probability.

Question

Fix $n \geq 2$. A uniformly random permutation $f: [n] \rightarrow [n]$ is being formed, where $[n] = \{1,2,\dots,n\}$. We say that $1 \leq i \leq n$ is fixed if $f(i) = i$. When $n = 10$, find the probability that $1$ is fixed, but $2$ is not fixed.

Practice focus

This Probability problem is tagged Combinations and Permutations, Conditional Probability. State the random variables and conditioning information explicitly, then check the result against boundary cases before opening hints or a solution.