# Number of integer solutions to $S_{1}+S_{2}+S_{3}+S_{4}+S_{5}+S_{6} = 30$ with constraints

Given the following equation

$$S_{1}+S_{2}+S_{3}+S_{4}+S_{5}+S_{6} = 30$$ such that $$S_{i} \in \mathbb{N} \cup \{0\} \space \forall i \in [6]$$

How many integer solutions exist, subject to the constraint:

$$\forall i \space S_{i}=i \pmod 3$$

Work so far:

Since $$S_{1}=10, S_2=2, S_3=3,S_4=4,S_5=5,S_6=6$$ is a solution, there is at least one solution. Also, without the constraint, the number of integral solutions is $${30 + 6-1 \choose 6-1} = {35 \choose 5}$$ therefore we know that the number $$n$$ of integral solutions under the constraint is $$1\leq n \leq {35 \choose 5}$$

Write $$S_i = 3a_i+i\pmod 3$$, so we have$$3a_1+1+3a_2+2+3a_3+0+3a_4+1+3a_5+2+3a_6=30$$
where $$a_i\geq 0$$, so $$a_1+a_2+...+a_6 = 8$$
and thus we have $${8+5\choose 5}$$ solutions