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Discrete mathematics is an area of study that deals with mathematical structures that are fundamentally discrete rather than continuous. In essence, it involves countable, distinct elements that do not require the notion of 'in between' values. This field encompasses a wide range of topics such as logic, set theory, graph theory, and combinatorics, which specifically deals with counting and arrangement problems.

Within discrete mathematics, permutations are a fundamental concept, particularly when it comes to arranging objects in specific orders. A permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement. For example, when looking at the different ways to arrange letters from a word like 'ALGORITHM', each unique sequence that can be made with the letters constitutes a different permutation. Understanding permutations is crucial in discrete mathematics as they are applied in various areas such as computer science algorithms, cryptographic systems, and even in organizing data structures.

Factorial notation is integral to combinatorics and permutations. The factorial of a non-negative integer, denoted by the exclamation mark (!), is the product of all positive integers less than or equal to that number. For instance, the factorial of 4, which is written as 4!, would be calculated as 4 × 3 × 2 × 1 = 24.

Factorials are particularly useful when determining the number of ways we can arrange a set number of objects. In the context of our problem with the word 'ALGORITHM', factorials help us calculate how many permutations exist depending on the number of letters we select. By using factorial notation, the complex process of counting each possible arrangement manually is eliminated, and we are provided with a powerful mathematical shortcut that simplifies computation significantly.

Combinatorics is a branch of mathematics focused on counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics, such as algebra, probability, and number theory, and has applications in diverse fields including computer science, statistics, and biology.

Counting the number of possible arrangements or selections is a primary concern in combinatorics. The problems in the exercise with the word 'ALGORITHM' demonstrate different combinatorial situations. From selecting and arranging a subset of letters without any constraints to doing so with specific positions being predetermined, we see the richness of counting techniques available. By applying the principles of combinatorics, we are able to swiftly navigate through complex scenarios, quantifying the possibilities with clear, logical steps and mathematical expressions.