Part of the effort that goes into the professionalism of mathematics is the elimination of ambiguity. One of the terms that mathematicians seem to take for granted is the permutation. While using the term permutation we, invariably, alternate between a relative and absolute sense of the word. The absolute sense is used with examples and in relation to combinations. Occasionally, the term arrangement is used in this way, but it is also used in the relative sense. For example, we classify permutation functions as dearrangements. Luckily we can clarify the situation by referring to the relative sense as a permutation function and the absolute sense as a fixed order.

These concepts are so important, but also so common that it is impractical to consistently distinguish between these two senses of permutation. I suggest that every introduction to abstract / modern algebra and combinatorics emphasize this distinction when defining permutations to better prepare students.