How to find the coefficient without a calculator? I wanted to solve this question without using a calculator.
Question: The number of non-negative integer solutions to
$$3x+y+z=24$$
By creating generating functions you have to find the coefficient of $x^{24}$ in the expression: $$\left(\frac{1}{1-x}\right)^{2}\left(\frac{1}{1-x^3}\right)$$
Using the theory I know about now, I would just split the problem into smaller parts adding all the combinations together while using the extended binomial theorem. But this takes a lot of time and I was wondering if there is a faster/easier way to find coefficients in terms of multiple generating functions by hand? If so, what are some recommended places to read about it?

Here's an alternative approach that uses stars and bars instead of generating functions.  Condition on the number of solutions with $x=k$, which reduces to the number of nonnegative integers solutions to $y+z=24-3k$:
\begin{align}
\sum_{k=0}^8 \binom{24-3k+2-1}{2-1}
&= \sum_{k=0}^8 (25-3k) \\
&= 9\cdot 25 - 3\sum_{k=0}^8 k \\
&= 9\cdot 25 - 3\cdot 9 \cdot \frac{0+8}{2} \\
&= 9(25 - 12) \\
&= 117
\end{align}
