What is the difference of an n-tuple and a permutation of n elements My understanding of
 n-tuple 
and
 a permutation of n elements 
is, that both are ordered sequences of n elements.
Are there differences in the objects correlating to these two terms ?
I guess it would be nice if any help could be coded in language than can be understood by someone fascinated in and swamped by mathematics, though any help is appreciated. 
 A: An $n$-tuple of elements in a set $X$ is a function $f:\{1,\ldots, n\}\rightarrow X$, whereas a permutation is a bijection $g:X\rightarrow X$.  The differences are that an $n$-tuple can have any codomain and need not be a bijection (or even an injection or surjection), while a permutation has both the domain and codomain equal and is a bijection, so they are very different objects in general.  
For example, we can represent an $n$-tuple as $(x_1,\ldots, x_n)$.  But because the $n$-tuple is any function as above, we could have $x_i=x_j$ for $i\neq j$ (i.e. not injective) and there is always an issue of surjectivity, as we could take the $n$-tuple with every entry being equal; if $X$ has more than one element, this won't be surjective.  On the other hand, a permutation of $\{1,\ldots, n\}$ is an $n$-tuple of elements in $\{1,\ldots, n\}$ such that we never have $x_i=x_j$ for $i\neq j$ (this ends up being sufficient for surjectivity as well).
However, in the case where we deal with permutations of finite sets, there is an instance in which they become similar.  When $X$ is a finite set, we can restrict to the case where $X=\{1,\ldots,n\}$ for some natural number $n$: then an $n$-tuple of elements of $X=\{1,\ldots, n\}$ is a function $f:\{1,\ldots, n\}\rightarrow \{1,\ldots, n\}$.  However, this still isn't necessarily a permutation, since we require that the function $f$ be a bijection.  Thus, you could think of permutations as special cases of $n$-tuples on a finite set of size $n$. 
A: A n-tuple is a sequence $(x_1, \cdots, x_n)$. A permutation of elements of a set $A$ is a bijection $\sigma:A \rightarrow A$, and for finite sets, we can represent it by matrices. For example, if $A = \{1,2,3,4\}$, then $$\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 2 & 3 & 1 & 4 \end{pmatrix}$$
represents $\sigma(1) = 2, \sigma(2) = 3, \sigma(3) = 1, \sigma(4) = 4$.
By heavily abusing the notation, we could think $\sigma(1,2,3,4) = (2,3,1,4)$, getting another n-tuple, but the objects are different. You can read more about this here.
