Editing Permutation (section)

===''k''-permutations of ''n''===
In older literature and elementary textbooks, a '''''k''-permutation of ''n''''' (sometimes called a '''[[Partial permutation#Restricted partial permutations|partial permutation]]''', '''sequence without repetition''', '''variation''', or '''arrangement''') means an ordered arrangement (list) of a ''k''-element subset of an ''n''-set.{{efn|More precisely, ''variations without repetition''. The term is still common in other languages and appears in modern English most often in translation.}}<ref>{{Cite web |title=Combinations and Permutations |url=https://www.mathsisfun.com/combinatorics/combinations-permutations.html |access-date=2020-09-10 |website=www.mathsisfun.com}}</ref><ref>{{Cite web |last=Weisstein |first=Eric W. |title=Permutation |url=https://mathworld.wolfram.com/Permutation.html |access-date=2020-09-10 |website=mathworld.wolfram.com |language=en}}</ref> The number of such ''k''-permutations (''k''-arrangements) of <math>n</math> is denoted variously by such symbols as <math>P^n_k</math>,  <math>_nP_k</math>, <math>^n\!P_k</math>, <math>P_{n,k}</math>, <math>P(n,k)</math>, or <math>A^k_n</math>,<ref>{{harvnb|Uspensky|1937|p=18}}</ref> computed by the formula:<ref>{{cite book|author=Charalambides, Ch A.|title=Enumerative Combinatorics|publisher=CRC Press|year=2002|isbn=978-1-58488-290-9|page=42|url=https://books.google.com/books?id=PDMGA-v5G54C&pg=PA42}}</ref>
: <math>P(n,k) = \underbrace{n\cdot(n-1)\cdot(n-2)\cdots(n-k+1)}_{k\ \mathrm{factors}}</math>,

which is 0 when {{math|''k'' &gt; ''n''}}, and otherwise is equal to
: <math>\frac{n!}{(n-k)!}.</math>

The product is well defined without the assumption that <math>n</math> is a non-negative integer, and is of importance outside combinatorics as well; it is known as the [[Pochhammer symbol]] <math>(n)_k</math> or as the <math>k</math>-th falling factorial power <math>n^{\underline k}</math>:<blockquote><math>P(n,k)={_n} P_k =(n)_k = n^{\underline{k}} .</math></blockquote>This usage of the term ''permutation'' is closely associated with the term ''[[combination]]'' to mean a subset. A ''k-combination'' of a set ''S'' is a ''k-''element subset of ''S'': the elements of a combination are not ordered. Ordering the ''k''-combinations of ''S'' in all possible ways produces the ''k''-permutations of ''S''. The number of ''k''-combinations of an ''n''-set, ''C''(''n'',''k''), is therefore related to the number of ''k''-permutations of ''n'' by:
: <math>C(n,k) = \frac{P(n,k)}{P(k,k)}= \frac{n^{\underline{k}}}{k!} =  \frac{n!}{(n-k)!\,k!}.</math>

These numbers are also known as [[binomial coefficient]]s, usually denoted <math>\tbinom{n}{k}</math>:<blockquote><math>C(n,k)={_n} C_k =\binom{n}{k} .</math></blockquote>