I was thinking that you first pick 1 of the 13 card types, and in that choose any two for the first pair. Then pick 1 of the 12 remaining card types and in that, choose any two for the second pair. Finally, pick 1 of the 11 remaining cards and in that, choose 1 card. Hence, the probability is:
$$ \frac{13\binom{4}{2} \cdot 12\binom{4}{2} \cdot 11\binom{4}{1}}{\binom{52}{5}} = 0.0951 $$
However, the solution is 0.0475 and it turns out I'm off by a factor of 2. Does anyone know why this method is not getting the right answer? I think it's something to do with double counting... But I can't seem to wrap my head around it.