Permutations
Definition: Permutations are a fundamental concept in combinatorics that deal with the arrangement of objects or elements in a specific order. A permutation is an ordered arrangement of objects taken from a set without repetition.
Key Concepts:

Permutation: A permutation is an ordered arrangement of "r" distinct objects selected from a set of "n" distinct objects without repetition. The order of arrangement matters in permutations.
The number of permutations of a set of $n$ objects is given by the following formula:
P(n) = n!
where $n!$ is the factorial of $n$, which is the product of all the positive integers less than or equal to $n$.
For example, the number of permutations of a set of 3 objects is:
P(3) = 3! = 3 * 2 * 1 = 6

Factorial Notation: Factorial notation is commonly used in permutations. The symbol "n!" represents the factorial of a nonnegative integer "n" and is defined as the product of all positive integers from 1 to "n."
n! = n × (n1) × (n2) × ... × 3 × 2 × 1

Permutation Formula: The number of permutations of "r" objects selected from a set of "n" distinct objects without repetition is calculated using the permutation formula:
P(n, r) = n! / (n  r)!
Where:
 P(n, r) represents the number of permutations of "r" objects from a set of "n."
 n! is the factorial of "n."
 (n  r)! is the factorial of "n  r."

Example: Suppose you have a set of 5 distinct books, and you want to arrange 3 of them on a shelf in a specific order. To find the number of permutations, use the formula:
P(5, 3) = 5! / (5  3)! P(5, 3) = 5! / 2! P(5, 3) = (5 × 4 × 3 × 2 × 1) / (2 × 1) P(5, 3) = 60 permutations
So, there are 60 different ways to arrange 3 books out of 5 on the shelf.

Permutation with Repetition: In some cases, you may need to calculate permutations with repetition. This occurs when objects can be repeated in the arrangement. The formula for permutations with repetition is:
P(n, r) = ${n}^{r}$
Where:
 P(n, r) represents the number of permutations of "r" objects from a set of "n" with repetition.
 n is the number of distinct objects.
 r is the number of positions or repetitions.

Application: Permutations are widely used in various fields, including mathematics, statistics, computer science, and cryptography. They are essential for solving problems related to arranging elements in specific orders, creating passwords, and analyzing data.

Circular Permutations: Circular permutations deal with arranging objects in a circle or cycle. In a circular permutation, the order of objects is considered, and it is one less than the number of linear permutations. The formula for circular permutations is:
P(n) = (n  1)!
Where:
 P(n) represents the number of circular permutations of "n" distinct objects arranged in a circle.