## Combinations with ripetition

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