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Discrete mathematics is a branch of mathematics primarily concerned with countable, distinct elements and structures, rather than continuous ones. It encompasses a wide range of topics, including logic, set theory, graph theory, and combinatorics, which is where permutations reside. Discrete mathematics lays the foundation for computer science and information theory, as it deals with algorithms, and data structures that are integral to these fields. Understanding how to manipulate and count distinct objects mathematically is crucial for solving many real-world problems.

In the context of our problem, discrete mathematics provides us with the tools to calculate the number of possible permutations, a concept that significantly contributes to areas such as probability, game theory, and cryptography.

Proof techniques in mathematics are methods used to establish the truth of propositions and theorems. Common techniques include direct proof, contradiction, induction, and combinatorial proofs, each suitable for different kinds of problems. Direct proof demonstrates that a statement follows logically from other already-proven statements or axioms. In contradiction, one assumes the opposite of what they want to prove, and shows this leads to an illogical or impossible scenario. Induction proves a base case and then shows that if it holds for an arbitrary case, it must hold for the next. Lastly, combinatorial proofs involve counting sets in two ways to show equality.

For proving our permutation formula, we deploy these techniques systematically, by first defining our operation—the permutation—and then manipulating this definition precisely, adhering to established mathematical principles to show that our equation holds true.

Permutations deal with the arrangements of a set of objects where order is important. In discrete mathematics, they are used to count the different ways to arrange a collection of distinct items. For example, if we consider the permutation of three distinct colors of balls - red, green, and blue, each unique arrangement, such as red-green-blue or blue-green-red, is a permutation.

In our exercise, we explore permutations that involve a fixed number of positions, captured by the notation P(n, k), where 'n' is the total number of items and 'k' is the positions to fill. This concept's fundamental in many fields, like statistics and operations research, where it helps strategize and deduce probabilities of certain outcomes arising from well-defined sets.

Factorial notation is a mathematical concept used to describe the product of an integer and all the non-zero integers below it. Denoted by the exclamation mark (!), the expression 'n!' is read as 'n factorial' and calculated as the product of all positive integers from 1 up to n. For instance, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

Factorials are deeply tied to permutations, as they represent the total number of ways 'n' objects can be arranged. A deeper look into our exercise shows how factorial notation simplifies the representation of permutations. In permutation formulae, factorials allow for the efficient expression of the number of arrangements. The concept of factorial is thus not just a fundamental part of the permutations but a widely used tool in combinatorics and various probability calculations.