# Combinations with ripetition

Hi,everyone .Let $A$ a no-empty set ,suppose $A$ is finite and let $n$,natural $\neq 0$, its cardinality ,let $k$ a natural number $\neq 0$: we say every ordinate $k$ -upla $( a_{1}...a_{k})$ is a disposition with repetition of class k of $A$ where $a_{i} \in A$.To define a combination with repetition of class $k$ of $A$ i gave the following deinition: let $( a_{1}...a_{k})$ , $( b_{1}...b_{k})$ two dispositions with repetition of class $k$ of $A$ .We say $( a_{1}...a_{k})$ and $( b_{1}...b_{k})$ are "of the same type" if $\{a_{1}...a_{k}\} =\{b_{1}...b_{k}\}$ , and ,set $\{a_{1}...a_{k}\}=\{c_{1}...c_{r}\}$ ,where r is the cardinality of $\{a_{1}...a_{k}\}$, we have $\forall i\in \{1,...,r\}$ $\sharp ( \{j \in\{1,...,k\}\mid a_{j}=c_{i} \} = \sharp ( \{j \in\{1,...,k\}\mid b_{j}=c_{i} \}$. So we can consider the following set $R= \{ (a,b) \in$ $\{$ every disposition with repetion of class k of $A \}^{2}\mid$ $a$ and $b$ are "of the same type" $\}$ and observe R is an equivalence relation on $\{$ every disposition with repetiton of class k of $A$} ,so we can say $\forall$ disposition with repetition of class k of $A$ } , $a$, the equivalence class of $a$ on $\{$ every disposition with repetition of class k of $A$ } modulo $R$ is a combination with repetition of class k of $A$.Is this definition already used ?