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The concept of a factorial is fundamental to permutations. It is a mathematical operation where you take an integer, referred to as 'n', and multiply it by all the positive integers below it down to one. This is denoted as 'n!'. For example, when asked to compute 5!, you would calculate it as follows:
\( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
Factorials are crucial when solving problems involving permutations because they represent the number of ways in which a set number of items can be arranged in sequence. When numbers get large, factorials can grow extremely fast, making manual calculation impracticable. In such scenarios, using a calculator or factorial tables is recommended to avoid errors.

In combinatorics, the arrangement of items is the core idea behind permutations. An arrangement is a specific order or sequence that a set of items is placed in. When dealing with permutations, each unique sequence counts as a different arrangement. The vital part to remember is that the position of each item matters.

For instance, if we have three books labeled A, B, and C, the sequence 'A-B-C' is a different arrangement from 'C-A-B'. Although it's the same set of books, their order changes the permutation. This concept extends to any group of distinct items, such as numbers, letters, or even colored balls, where each unique ordering of the items is counted as a distinct permutation.

What distinguishes permutations from combinations is the 'order of arrangement'. It is critical to understand that in permutations, changing the order changes the entire configuration, essentially creating a new permutation.

For example, if we're looking at a simple case like arranging two letters, say 'X' and 'Y', there are two possible permutations: 'XY' and 'YX'. Despite having the same elements, switching their order results in different arrangements and thus counts separately when calculating permutations. This principle applies to larger sets of items too, as seen in the exercise with the 11 distinct objects - every unique sequence generated from rearranging those objects counts as its own permutation.