---
kind: statement
msc:
- 05A05
name: Permutations fixing a given set of points
role: lemma
sources:
- locator: source/s_inclusion-exclusion_derangements.ptx:42
  url: https://github.com/mitchkeller/applied-combinatorics/blob/ebd95d6bd59e23c633c65cf560bcb93efa912aae/source/s_inclusion-exclusion_derangements.ptx#L42
tag: N8YF97HHQ5
uses:
- 0BQMWCNS8S
- 0TH61JR04G
---

## Statement

Let $n$ be a natural number, and let $S$ be a subset of $[n]=\{1,\dots,n\}$.
Let $N(S)$ be the number of [[permutations|0BQMWCNS8S]] of $[n]$ that fix every element of $S$.
Then $N(S)$ depends only on the size of $S$.
If $|S|=k$, then $N(S)=(n-k)!$.

## Remarks

A permutation that fixes each point of $S$ is determined by its values on the other $n-k$ points.
These values form an arbitrary permutation of the $n-k$ remaining points.
This count is the input for the inclusion-exclusion count of [[derangements|0TH61JR04G]].