AMC 10 2021-fall-a Problem 21

Each of the $20$ balls is tossed independently and at random into one of the $5$ bins. Let $p$ be the probability that some bin ends up with $3$ balls, another with $5$ balls, and the other three with $4$ balls each. Let $q$ be the probability that every bin ends up with $4$ balls. What is $\frac{p}{q}$? $\textbf{(A)}\ 1 \textbf{(B)}\ 4 \textbf{(C)}\ 8 \textbf{(D)}\ 12 \textbf{(E)}\ 16$