We want to find the probability of being dealt exactly one pair when we are dealt  5 cards.

The $\displaystyle \binom{13}{1}$ chooses the rank of the pair, and the $\displaystyle\binom{12}{3}$ chooses three other distinct ranks so that we have one and only one pair. Then $\displaystyle \binom{4}{2}$ chooses which two suits will be chosen for the pair, and $\displaystyle \binom{4}{1}^3$ chooses the suit for each of the single cards.

Thus, the chances of getting exactly one pair are $$\dfrac{\displaystyle \binom{13}{1}\binom{12}{3}\binom{4}{2}\binom{4}{1}^3}{\displaystyle\binom{52}{5}}.$$