Odds against a four of a kind in five cards

Five cards are dealt from a standard $52$-card deck. If $p$ is the probability that four of them share a rank, what is $1/p$?

$4165$. There are $13$ choices of rank for the four matching cards and $48$ choices for the fifth card, giving $624$ hands out of $\binom{52}5=2{,}598{,}960$. The ratio is $\tfrac1{4165}$.