Problem:

Determine the number of 8-tuples of nonnegative integers $\left(a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4}\right)$ satisfying $0 \leq a_{k} \leq k$, for each $k=1,2,3,4$, and
$$a_{1} + a_{2} + a_{3} + a_{4} + 2 b_{1} + 3 b_{2} + 4 b_{3} + 5 b_{4} = 19.$$