Chance a five-card hand is exactly two pair

Five cards are dealt from a well-shuffled standard 52-card deck. Find the probability that the hand is two pair: two cards of one rank, two cards of a second rank, and a fifth card of a third rank.

$\tfrac{198}{4165}\approx 0.0475$. Pick the two paired ranks in $\binom{13}{2}$ ways and two suits for each in $\binom42^2$ ways, then the kicker from the $44$ cards of other ranks. Dividing $\binom{13}{2}\cdot 36\cdot 44=123{,}552$ by $\binom{52}{5}$ gives $\tfrac{198}{4165}$.