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The concept of a factorial is used to count the number of ways we can arrange different objects without repetition. In mathematics, the factorial of a non-negative integer is denoted by and is the product of all positive integers less than or equal to . For instance, (pronounced 'four factorial') is calculated as .

Factorials play a significant role in various areas of mathematics, including algebra and combinatorics, particularly in permutations and combinations. It is the cornerstone for understanding how many different ways we can arrange or choose items. Recognizing that factorials can grow very large very quickly is essential because this explosion in size has implications in calculating probabilities and combinations where factorials are used.

An arrangement of elements refers to the different ways objects can be ordered or structured. When considering permutations, every unique order of a set of elements is a distinct arrangement. It's essential to determine whether repetition is allowed because this affects the total number of possible arrangements.

The task of writing all permutations of the letters A, B, C, and D implies finding every possible combination of these letters in a four-letter sequence. By arranging these letters in every conceivable sequence where no letter is used more than once, we can explore the practical application of permutations. The complexity of possible arrangements increases with the number of elements, which is why understanding the systematic approach to listing permutations is beneficial. In the example given, starting with a fixed letter and rotating the others systematically ensures all permutations are covered without repetition.

Combinatorics is a field of mathematics focused on counting, arrangement, and combination of elements within a set according to specific rules. It's heavily applied in fields like probability theory, statistics, and computer science. Within combinatorics, there are various concepts including permutations, combinations, and the binomial theorem.

Permutation is a key concept where the order of arrangement matters, as seen with the letters A, B, C, and D. Another important concept in combinatorics is combinations, where the order does not matter. Understanding the fundamental differences between these concepts is vital in solving problems related to counting and arrangement. In combinatorics, factorial notation often arises as a way to simplify expression of large numbers that appear in formulae for counting permutations and combinations.