Ratio of a 3-5-4-4 split to an even split

Sixteen distinguishable cherries are each dropped independently and uniformly into one of $4$ distinguishable bowls. Let $a$ be the probability that the bowl counts are $3$, $5$, $4$, $4$ in some arrangement (one bowl with $3$, one with $5$, two with $4$), and let $b$ be the probability that every bowl gets exactly $4$. Find $a/b$.

$48/5$. For a fixed assignment of counts to bowls the probability is a multinomial coefficient over $4^{16}$. The $3,5,4,4$ pattern can be placed on the bowls in $4!/2!=12$ ways, so $a/b=12\cdot\tfrac{4!\,4!}{3!\,5!}=12\cdot\tfrac45=\tfrac{48}{5}$.