Counting integer quadruples

Think of the total as stars separated by dividers.

1. Arrange $6$ stars and $3$ dividers in a row.
2. An arrangement is fixed by the positions of the dividers among $9$ slots, giving $\left(\genfrac{}{}{0ex}{}{9}{3}\right)$.
3. Compute $\left(\genfrac{}{}{0ex}{}{9}{3}\right)=84$.
4. The count is $84$.

Answer: $84$