Combinatorics: terminology for permutations and combinations I find it very difficult to understand the correct translation of these terms. So far I know this (translation Italian - English) given $n$ number of items and $k$ number of groups:


*

*Combinazioni (it) = Combinations (en)


Here the order doesn't matter and we calculate the result with $$\frac{n!}{ k!(n-k)!}$$ 
In the case of no repeated items, instead we use 
$$\frac{(n+k-1)!}{k!(n-1)!}$$

So far so good but now I have trouble with permutations. My Italian probability university book gives me these $2$ definitions:


*

*Permutazioni (it) = Permutations (en)


Here the order is important and $n \neq k$. We use $n!$ when there aren't repeated items and 
$$\frac{n!}{x_1!x_2!x_3!\ldots x_k!}$$ 
when there are repetitions.


*

*Disposizioni (it) = ??? (en)


I have googled this a lot but I cannot find a proper translation. Here the order is important and $n = k$. We use 
$$\frac{n!}{(n-k)!}$$ 
when there aren't repeated items and $n^k$ when there are repetitions. 
Is there a proper translation for this?

Note: from my researches I have seen that they are repetitions as well and there isn't a proper definition. My sources aren't a lot but I see this:


*

*Disposizioni (italian)

*Permutations (english)


I have made a C++ app that involves combinatorics so I need to give a name at the "Disposizioni" group. Any help?
 A: Combinations or $k$-Combinations:  The number of ways of making an unordered selection of $k$ objects from a set of $n$ objects.  Alternatively, the number of $k$-element subsets of an $n$ element set.
$$\binom{n}{k} = \frac{n!}{k!(n - k)!}$$
Combinations with repetition:  The number of ways of making an unordered selection of $k$ objects from a set of $n$ types of objects when repetition is permitted.  
$$\binom{k + n - 1}{n - 1}$$
The formula you gave is the number of solutions of the Diophantine equation
$$x_1 + x_2 + x_3 + \cdots + x_n = k$$
in the nonnegative integers.
Permutations:  The number of ways of arranging $n$ distinct objects in order.
$$n!$$
The expression $n!$ is read "$n$ factorial''.
Permutations of a multiset:  Given a multiset 
$$\{n_1 \cdot x_1, n_2 \cdot x_2, n_3 \cdot x_3, \ldots, n_k \cdot x_k\}$$
where $n_j$ is the multiplicity of the element $x_j$, $1 \leq j \leq k$, and $n = n_1 + n_2 + n_3 + \cdots + n_k$ is the number of elements in the multiset, the number of distinguishable arrangements of the objects in the multiset is
$$\binom{n}{n_1, n_2, n_3, \ldots, n_k} = \frac{n!}{n_1!n_2!n_3! \ldots n_k!}$$
Permutations or $k$-Permutations:  The number of ways of making an ordered selection of $k$ objects selected from a set of $n$ objects.
$$P(n, k) = \frac{n!}{(n -k)!}$$
Note that $P(n, n) = n!$.
Permutations with repetition:  The number of ways of sequences of length $k$ that be formed from a set with $n$ elements when repetition is permitted.
$$n^k$$
