I have $10$ indistinguishable cars , $12$ indistinguishable balls, $14$ indistinguishable teddy bears.I want to distribute them to $3$ different children in a kindergarten such that each child will take exactly $7$ toys.How many distributions are there?
$\mathbf{\text{My attempt:}}$ First of all, I thought that it can be solved by generating functions easily. However, the process became suddenly cumbersome.
If each child will take exactly $7$ toys , then we need $21$ toys in total. For example , $10$ cars, $6$ balls and $5$ teddy bears can be a sample. By using this information, I decided to use generating functions such that if a child take exactly $7$ toys, then his generating function form is $$(x^7+y^7+z^7+x^6y+x^6z+x^5y^2+x^5yz+...+yz^6)$$ where $x,y,z$ represents cars, balls and teddy bears, respectively. As you count by combination with repetition, there are $\binom{7+3-1}{7}=36$ terms in the tuple.
Because of there are $3$ children , we will deal with $$(x^7+y^7+z^7+x^6y+x^6z+x^5y^2+x^5yz+...+yz^6)^3$$
However, there is a problem for me. Calculating the coefficients for each possible selection is very cumbersome. For example, we said that one of the possible toy selections of $21$ of $36$ is $10$ cars, $6$ balls and $5$ teddy bears. In that sample, we should find $$[x^{10}y^6z^5](x^7+y^7+z^7+x^6y+x^6z+x^5y^2+x^5yz+...+yz^6)^3$$
As you realize there are many other toy selections out of $36$, for example $5$ cars, $8$ balls and $8$ teddy bears is an another sample.
I want you guys to handle this cumbersome selection process. How can I find the all coefficients of this generating function for all possibilities?
Please do not suggest using a computer algorithm. I am here to see a mathematical approach I could apply to my problem. What's more, if you have another approach, feel free and share it with me. I am open to another methods.