jmc

We use casework: Alpha purchases $3$ items, Beta $0$. Then there are $\left(\genfrac{}{}{0ex}{}{8}{3}\right)=56$ ways for Alpha to pick $3$ different items. Alpha purchases $2$ items, Beta $1$. Then there are $\left(\genfrac{}{}{0ex}{}{8}{2}\right)=28$ ways for Alpha to pick $2$ different items and there are $5$ ways for Beta to choose her oreo, with a total of $28\cdot 5=140$. Alpha purchases $1$ items, Beta $2$. Then there are $8$ ways for Alpha to pick $1$ different items. There are $\left(\genfrac{}{}{0ex}{}{5}{2}\right)=10$ ways for Beta to choose two distinct oreos, and $5$ ways for Beta to choose two oreos of the same flavor. This totals to $8\cdot (10+5)=120$. Alpha purchases $0$ items, Beta $3$. There are $\left(\genfrac{}{}{0ex}{}{5}{3}\right)=10$ ways for Beta to choose three distinct oreos. For Beta to choose two oreos of the same flavor and another oreo, there are $5$ choices for the first and $4$ choices for the second, with a total of $20$ choices. There are $5$ ways to choose three of the same flavored oreo. This totals to $10+20+5=35$. The total is $56+140+120+35=\boxed{351}$.