"Assigning to teams" as a function

Question

Given sets $A=\{1,2,3,4,5,6,7,8,9\}$ and $B=\{r,g,b\}$, with $r$, $g$ and $b$ teams (red, green and blue) of sizes $2$, $3$ and $4$, respectively, can we use function notation to express the idea of assigning a given number of elements from $A$ to each team?

Answer 1

Here is a variant which might fit the needs.

Let $A=\{1,2,3,4,5,6,7,8,9\}$ and $B=\{r,g,b\}$. We consider a function $f$ which fulfills \begin{align*} &f:A\to B\\ &|f^{-1}(r)|=2, |f^{-1}(g)|=3, |f^{-1}(b)|=4\tag{1}\\ \end{align*}

• Since $f$ is a function from $A$ to $B$ it has domain $A$ and each of the nine elements of $A$ is mapped to $B$.

• The pre-images of $\{r,g,b\}$ have according to (1) a size of $2,3$ and $4$ giving a sum of $2+3+4=9$. Since each of the nine elements of $A$ has to be addressed, (1) implies we have a partitioning of $A$.

So we describe with $f$ one out of $\binom{9}{2}\binom{7}{3}\binom{4}{4}=1\,260$ functions from $A$ to $B$ which fulfill (1).