Chance of a full house in five cards

Five cards are dealt from a well-shuffled standard $52$-card deck. What is the probability that they form a full house, i.e. three cards of one rank plus two cards of another rank?

$6/4165$ (about $0.144\%$). Choose the rank and suits of the triple, $13\binom43$, then the rank and suits of the pair, $12\binom42$, giving $3744$ hands out of $\binom{52}{5}=2{,}598{,}960$.