Number of strings of distinct letters

Builds on Falling factorial Number of injections

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 $(m)_n$ , 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.

Reading path

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Cardinality of a finite set definition
Falling factorial definition
Number of injections statement
Number of strings of distinct letters statement

Tag	`W0J0GT3PEF`
URI	`m2://W0J0GT3PEF`
Kind	statement, theorem
Writing	W0 stub
MSC	05A05
Status	This copy of the site has no build data. Run `m2 status W0J0GT3PEF`.
Uses	Falling factorial, Number of injections
Changed	4 November 2026

Number of strings of distinct letters

Statement

Remarks

Reading path

Further reading

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