Additionally, at least one but not more than 5 boxes of each type are chosen? I was told to solve i.t.o boxes.
I know this question was asked before but none of the answers helped and no work was shown. Here's my thought process..
If I think in terms of boxes, I let $e_i$ be the number of boxes I picked from the $ith$ type. I will have a total of $15$ boxes. I know that I need to pick $1$ box of each kind and no more than $5$. That leaves me with the following: $e_1 + e_2 + e_3 + e_4 + e_5+ e_6 + e_7 = r$ where $e_i$ ranges from $1$ to $5$. Translating it into an integer equation I have the following: $g(x)= (x^1+x^2+x^3+x^4+x^5)^7$, I also know that I need to figure out the coefficient of $x$ raised to the $15$. Now I can try to calculate all the formal products but this seems rather inefficient ..if that is the only way to solve, what's a good method/way to verify I have all possible formal products. Thank you in advance.