---
kind: statement
msc:
- 05A05
name: Number of strings of distinct letters
role: theorem
sources:
- locator: source/s_strings_permutations.ptx:60
  url: https://github.com/mitchkeller/applied-combinatorics/blob/ebd95d6bd59e23c633c65cf560bcb93efa912aae/source/s_strings_permutations.ptx#L60
tag: W0J0GT3PEF
uses:
- 0N5M924N3P
- 0X0N833FF8
---

## Statement

Let $X$ be a set with $m$ elements, and let $n$ be a positive integer with $n \le m$.
Call a string over $X$ a permutation if no letter occurs twice in it.
Then the number of such strings of length $n$ is $P(m,n) = m(m-1)\cdots(m-n+1)$.

## Remarks

Here $P(m,n)$ is the [[falling factorial|0N5M924N3P]] $(m)_n$, which equals $m!/(m-n)!$.
A string of length $n$ with distinct letters is the same as an injective function from $\{1,\dots,n\}$ to $X$.
So the count is a special case of the [[number of injections|0X0N833FF8]].