Finding the number of nonnegative integer solutions of $z_1 + z_2 + z_3 + z_4 = 10$ satisfying given restrictions

Question

Find the number of solutions of $$z_1 + z_2 + z_3 +z_4=10$$ where $z_i$ are nonnegative integers such that $z_1\le 4,z_2$ is odd,$z_3$ is prime and $z_4\in\{1,2,3,6,8\}$

Answer 1

The restricions are so haphazard that there is nothing else but going through the cases. I let Mathematica do it for me:

The answer then is $24$.