Probability of exactly one pair in five-card poker

Five cards are dealt from a well-shuffled standard $52$-card deck. Find the probability that the hand is exactly one pair: two cards share a rank and the remaining three cards have three other ranks, all different from each other.

$\tfrac{352}{833}\approx0.423$. Choose the paired rank and its suits, $13\binom42$, then three other distinct ranks with any suits, $\binom{12}{3}4^3$. Dividing $13\cdot6\cdot220\cdot64=1{,}098{,}240$ by $\binom{52}{5}=2{,}598{,}960$ gives $\tfrac{352}{833}$.