imc

We don't care about which books Harold selects. We just care that Betty picks $2$ books from Harold's list and $3$ that aren't on Harold's list. The total amount of combinations of books that Betty can select is $\left(\genfrac{}{}{0ex}{}{10}{5}\right)=252$. There are $\left(\genfrac{}{}{0ex}{}{5}{2}\right)=10$ ways for Betty to choose $2$ of the books that are on Harold's list. From the remaining $5$ books that aren't on Harold's list, there are $\left(\genfrac{}{}{0ex}{}{5}{3}\right)=10$ ways to choose $3$ of them. $\frac{10\cdot 10}{252}=\boxed{\frac{25}{63}}$