Difference between arrangement and order

Question

I am little bit confused about the difference between the words arrangement and order. At first, I thought the two words were the same, but then I came up with an example which made me think that these two words are different. The example was "Arrange the following numbers in ascending order" Please clearly explain the meanings of the two words with examples.

Answer 1

An arrangement is more general than an ordering: an arrangement of objects can be made in any way you like. Flowers can be arranged, but in general are not ordered.

An ordering is a specified arrangement. You can order ascending, descending, alphanumerically or any other way you like so long as it is clearly (and ideally unambiguously) stated.

So, $(123)$ is an arrangement of the first three integers, which is coincidentally an ascending ordering. $(123),(132),(231),(213),(312),(321)$ are all arrangements of the first three integers, with no specific ordering for them.

You may also come across the term permutation which is a rearrangement of a given sequence, again usually without a specified ordering.

Answer 2

Arrange and order both have several meanings, but in your example arrange is a verb meaning ‘to put a group of objects in a particular sequence’, and order is a noun meaning ‘a sequence of one thing following another’. More generally, to arrange a collection of things is to put them in particular positions relative to one another, and possibly in particular orientations as well. One can, for instance, arrange flowers in a vase to make an attractive display or arrange the contents of one’s refrigerator to maximize the available space.

Those examples don’t involve order, but many do. For instance, one can arrange a group of children in order of increasing height, meaning that the shortest child comes first and is followed by increasingly taller children, with the tallest child last. One can arrange books on a shelf in alphabetical order by author’s last name, so that books by Asimov come before books by Bester, which come before books by Clarke, and so on all the down to books by Zelazny at the end. In the first of these examples the sequence is determined by height; in the second, by the standardized linear sequence of the alphabet. In the example in the question the sequence is determined by the natural order of the numbers being placed in a linear sequence.