In this problem, I found that the answer is the coefficient of $x^{46}$ in $\left(\displaystyle\sum_{r=0}^{3} x^r \right)^6\left(\displaystyle\sum_{r=0}^{8} x^r \right)^4$
Is there possibly a way to find it manually, without using a CAS?
Well, you could use a CAS to check whether the coefficient is indeed $709$, the answer to the original problem.
Basically you just need to adapt my answer to the initial thread.
The maximum power in this polynomial is 50. So if you want to find 46, you need to take the maximum power in almost all factors (1+x+x² is a factor) except in a limited number. If this "limited number" is 4, then you have "10 choose 4" possibilities and in each of these 4 factors, instead of taking the maximum power, you take x² or x^7. If this limited number is 3, then you have 3*(10 choose 3) since you need to choose the 3 factors and then you have the choice between x^(n-1)/x^(n-1)/x^(n-2)... ...
Clearly, this solution will not satisfy you I guess because it's not formalized but it can be formalized.
PS : i'm sorry i cannot just comment because my reputation is too low.