Arrangements

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Arrangements of $k$ [distinct] objects chosen among $n$ [distinct] objects are permutations (tuples) of $k$ [distinct] elements taken from sets of $n$ [distinct] elements.

For example, the arrangements of 2 [distinct] elements taken from the set of 3 [distinct] elements $\{i,j,k\}$ are

$\{(i,j),(i,k),(j,i),(j,k),(k,i),(k,j)\},$

where $(i,j)$ denotes a tuple.

Number of arrangements

The number of arrangements (permutations) of $k$ [distinct] elements taken from sets of $n$ [distinct] elements is given by

$P_{k}^{n}=k!\,C_{k}^{n}=k!{n \choose k}=k!\,{\frac {n!}{k!\,(n-k)!}}={\frac {n!}{(n-k)!}},\,$

where $\scriptstyle C_{k}^{n}={n \choose k}$ is the number of combinations of $k$ [distinct] elements taken from sets of $n$ [distinct] elements, $\scriptstyle {n \choose k}\,$ are binomial coefficients and $\scriptstyle n!\,$ is the factorial of n.

The number of arrangements of any $k$ [distinct] elements taken from sets of $n$ [distinct] elements is given by

$a_{n}=\sum _{k=0}^{n}P_{k}^{n}=\sum _{k=0}^{n}{\frac {n!}{(n-k)!}}=n!\sum _{k=0}^{n}{\frac {1}{(n-k)!}}=n!\sum _{k=0}^{n}{\frac {1}{k!}}.\,$

See also

• Permutations
• Derangements
• Combinations