In how many ways can I collect a total of $20$ dollars from $4$ different children and $3$ different adults, if each child can contribute up to $6$ dollars, each adult can give up to $10$ dollars, and each individual gives a nonnegative whole number of dollars?
How should I start? Generating functions? Counting?
Quite likely generating functions have a tidy solution. Otherwise you have to use a bit of inclusion-exclusion:
$$\begin{align} {26 \choose 6} - 4{19 \choose 6} &- 3{15 \choose 6} + {4 \choose 2}{12 \choose 6} + 4\cdot 3 {8 \choose 6} \\[1em] &= 230230-108528-15015+5544+336 \\ &= 112567 \end{align}$$
representing options without limits, less options where one child or one adult breaks their limit, adding back 2 children or an adult/child combination breaking their limits. You can't get three donors breaking their limit simultaneously.