## Permutation

In mathematics, a permutation is a rearrangement of a set of objects in a specific order. Permutations are used to count the number of ways that a set of objects can be arranged. A permutation of a set S is a one-to-one and onto mapping of S onto itself.

More formally, a permutation of a set S is a bijective function \( \sigma \): \(S \rightarrow S \) . In other words, \(\sigma \) (Sigma) is a function that maps every element in S to a unique element in S, and every element in S is mapped to exactly once. We can represent a permutation \( \sigma \) of a set S by writing down its values in a particular order, for example:

\( \sigma = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 3 & 1 & 4 & 2 \end{pmatrix} \)

This notation means that \( \sigma(1) = 3 \), \( \sigma(2)=1 \), \(\sigma(3)=4 \) and \( \sigma(4)=2 \). In other words, the first row represents the elements of S in their original order, and the second row represents their order after applying the permutation \( \sigma \).

The number of permutations of a set S with \(n\) elements is denoted by \(n!\), which is read as "n factorial". The factorial function is defined as:

\( n!=n \cdot (n-1) \cdot (n-2) \ldots 2 \cdot 1 \)

For example, \(5!=5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=120 \) , which means that there are 120 permutations of a set with 5 elements.

Permutations can be used to solve various counting problems. For example, suppose we have 5 different books and we want to arrange them on a shelf. The number of ways to arrange the books is given by the number of permutations of a set with 5 elements, which is \( 5!=120 \).

Another example is the number of ways to select a committee of 3 people from a group of 10 people. The number of ways to select the committee is given by the number of permutations of a set with 10 elements taken 3 at a time, which is denoted by \(_{10} P_3 \) and is calculated as:

\( _{10} P_3 =\frac{10!}{(10-3)!} = 10 \cdot 9 \cdot 8 =720 \)

In general, the number of permutations of a set with \(n\) elements taken \(r\) at a time is denoted by \(_n P_r\) and is calculated as:

\( _n P_r= \frac{n!}{(n-r)!} \)

In conclusion, permutations are a fundamental concept in combinatorics and are used to count the number of ways that a set of objects can be arranged. The number of permutations of a set with n elements is \(n!\), and the number of permutations of \(r\) elements taken from a set of \(n\) elements is \(_n P_r\).