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Combinatorics is a branch of mathematics that deals with the counting, arrangement, and combination of objects. It plays a crucial role in various fields such as mathematics, computer science, and statistics.

Think of combinatorics as the science of counting without actually having to count every possible option. This is handy because, in many scenarios, directly counting each configuration can be impractical or even impossible due to the sheer number of possibilities. So, combinatorics provides us with shortcuts through formulas and principles that help in systematic counting.

One of the most fundamental concepts within combinatorics is the idea of a 'permutation', which refers specifically to arrangements where the order is important. Understanding permutations is essential for solving problems related to password generation, seating arrangements, scheduling tasks, and organizing tournaments, among many others.

The order of arrangement plays a pivotal role when it comes to permutations. It can transform one set of objects into a different configuration simply by changing the positions of those objects relative to each other.

To illustrate, consider a simple lock with a combination. If the correct order for the lock is '123', then '321' or '213' will not work, even though they contain the same numbers. This shows how the order can change the identity and function of an arrangement.

In mathematical terms, each unique sequence derived from a set of objects by varying their order constitutes a different permutation. This is why 'AB' is distinct from 'BA'; the same elements are present, but the order changes their arrangement, thus creating two separate permutations.

Mathematical concepts are the underlying principles that form the building blocks for all mathematical theory and applications. These include notions such as numbers, shapes, functions, and, in our context, permutations. Learning and understanding these concepts can seem abstract but are integral in applying them to real-world problems.

In the case of permutations, the underlying mathematical concept is centered around arranging objects in a specific order. Using permutations involves utilizing formulas like the permutation formula, which calculates the number of possible arrangements for a given number of objects. The formula is expressed as \( n!/(n-r)! \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to arrange.

The takeaway is that mathematical concepts like permutations allow us to model and solve problems more efficiently. They show us patterns in complex situations and can be applied across different scenarios, underscoring the flexibility and power of mathematics.