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permutation

A permutation is a rearrangement of the elements of a set into a new order. Equivalently, it is a bijection from the set to itself that reassigns each element to a unique position. For a finite set with n elements, the total number of distinct permutations is n!, the product of all positive integers up to n.

When the elements are distinct, permutations count all possible orderings of those elements. For example, the

In algebra, the set of all permutations of n elements forms the symmetric group S_n under composition

If repetitions are allowed, or the set contains repeated elements (a multiset), the number of distinct permutations

Applications of permutations appear in counting problems, sorting and searching algorithms, cryptography, and the study of

three-element
set
{A,
B,
C}
has
six
permutations:
ABC,
ACB,
BAC,
BCA,
CAB,
CBA.
The
concept
generalizes
to
any
n-element
set,
including
the
numbers
1
through
n
or
any
collection
of
distinct
symbols.
of
functions.
Any
permutation
can
be
expressed
as
a
product
of
disjoint
cycles,
and
it
can
also
be
written
as
a
product
of
transpositions
(swaps
of
two
elements).
The
parity
of
a
permutation—whether
it
is
an
even
or
odd
permutation—depends
on
the
parity
of
the
number
of
transpositions
in
any
such
decomposition.
is
reduced
to
n!
divided
by
the
product
of
the
factorials
of
the
multiplicities
of
each
repeated
element.
symmetry
in
algebraic
structures.