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Permutations revolve around the concept of arranging objects in a specific order. The different ways in which a set of objects can be ordered are what determine the permutations of the set. For example, if we're looking at a simple set of objects—say, the first three letters of the alphabet—their permutations would include six arrangements like ABC, ACB, BAC, BCA, CAB, and CBA.

Understanding permutations is crucial in various fields such as mathematics, computer science, and even in day-to-day activities where the order of things matters. Combinatorics, an area of mathematics, focuses extensively on counting permutations, especially when it comes to complex scenarios with a large number of objects, where manual counting isn't feasible.

The concept of factorial is integral to combinatorics and permutations. The factorial of a number, expressed as 'n!', represents the product of all positive integers from 1 up to that number. For example, the factorial of 4 (denoted as 4!) would be calculated as:
\[ 4! = 4 \times 3 \times 2 \times 1 = 24. \]
It's worth noting that by definition, the factorial of zero is 1, or \(0! = 1\), which may seem counterintuitive but is essential in various mathematical proofs and helps to simplify many combinatorial formulas.

Combinatorics is the branch of mathematics focused on the study of finite or countable discrete structures. In essence, it's about counting, arranging, and finding patterns in sets of elements. Within combinatorics, the concepts of permutations and combinations are two sides of a coin; while permutations care about the order of the elements, combinations are concerned with the selection of elements irrespective of the order.

Application in Real Life

Imagine organizing books on a shelf or creating passwords from a set of characters; combinatorics provides the tools to count the number of possible arrangements or selections. This branch is not just theoretical—it has practical applications in areas like cryptography, computer algorithms, and probability theory.

Mathematical identities are equations that are universally true for all allowed values of the variables within them. They are a fundamental part of mathematics, providing a way to transform complex problems into simpler forms that we can more easily solve or understand.

In the context of permutation identity, as posed in the exercise, it simplifies the understanding of how the number of permutations changes—or in this case, doesn't change—when we are permuting an entire set versus all but one element of it. By recognizing that \(_{n}P_{n}=n!\) and \(_{n}P_{n-1}=\frac{n!}{1!}=n!\), we leverage this identity to showcase equalities that might not be immediately apparent but have wide-reaching implications in mathematical theory and practice.