smc

For simplicity purposes, we assume that the balls and the bins are both distinguishable. Recall that there are $5^{20}$ ways to distribute $20$ balls into $5$ bins. For $p,$ we choose one of the $5$ bins to have $3$ balls and another one of the $4$ bins to have $5$ balls. We get $$p=\frac{5\cdot 4\cdot \left(\genfrac{}{}{0ex}{}{20}{3,5,4,4,4}\right)}{5^{20}}\text{ and }q=\frac{\left(\genfrac{}{}{0ex}{}{20}{4,4,4,4,4}\right)}{5^{20}}.$$ Therefore, the answer is $$\frac{p}{q}=\frac{5\cdot 4\cdot \left(\genfrac{}{}{0ex}{}{20}{3,5,4,4,4}\right)}{\left(\genfrac{}{}{0ex}{}{20}{4,4,4,4,4}\right)}=\frac{5\cdot 4\cdot \frac{20!}{3!\cdot 5!\cdot 4!\cdot 4!\cdot 4!}}{\frac{20!}{4!\cdot 4!\cdot 4!\cdot 4!\cdot 4!}}=\frac{5\cdot 4\cdot (4!\cdot 4!\cdot 4!\cdot 4!\cdot 4!)}{3!\cdot 5!\cdot 4!\cdot 4!\cdot 4!}=\frac{5\cdot 4\cdot 4}{5}=\boxed{16}.$$