\(|A| = {8 \choose 2}\text{.}\) First give Alberto 5 cookies, then distribute the remaining 6 to the three kids without restrictions, using 6 stars and 2 bars.
\(|B| = {8 \choose 2}\text{.}\) Just like above, only now Bernadette gets 5 cookies at the start.
\(|C| = {8 \choose 2}\text{.}\) Carlos gets 5 cookies first.
\(|A \cap B| = {3 \choose 2}\text{.}\) Give Alberto and Bernadette 5 cookies each, leaving 1 (star) to distribute to the three kids (2 bars).
\(|A \cap C| = {3 \choose 2}\text{.}\) Alberto and Carlos get 5 cookies first.
\(|B \cap C| = {3 \choose 2}\text{.}\) Bernadette and Carlos get 5 cookies first.
\(|A \cap B \cap C| = 0\text{.}\) It is not possible for all three kids to get 4 or more cookies.