I am trying to understand the formal definitions of a permutation. This one is from Wikipedia:
In mathematics, the notion of permutation relates to the act of arranging all the members of a set into some sequence or order
A sequence is an enumeration of objects with repetitions allowed. When I am permuting a set (this already implies that all objects in it are unique), none of its permutations can have repeated elements. So why is it defined as a sequence?
One of the other definitions is that a permutation is a sequence that contains all elements of the set exactly once.
Why is there again an emphasis on calling it a sequence?
Why not just call a permutation a set?
Also, permutations can be made from a subset of the original set, ie. I can take 3 elements from a set at a time that contains 5 elements and then permute them. So, the resultant permutations will not contain all the elements of the set exactly once. So how does the second definition hold up there?
Please note that I do not have a pure mathematics background, so explanations in "laymans terms" would be greatly appreciated.