How many 5-card hands contain two of one rank and three of another rank? What is the probability of being dealt such a hand?

I am not actually taking any classes that cover probability at the moment, so I don't have a lot of background knowledge for this one. I am, however, covering inclusion/exclusion at the moment so I gave it a shot.

There are 13 ranks and I need to choose 2:

$\displaystyle {13\choose 2}$

Once this has been done, I need to choose 2 from one rank:

$\displaystyle {4\choose 2}$

and 3 from the other rank:

$\displaystyle {4\choose 3}$

So there are:

$\displaystyle {13\choose 2}\times {4\choose 2}\times {4\choose 3}$ possible hands?

My answer is different from the one provided, I'm not sure why...

How do I calculate the probability of being dealt such a hand once I know how many possibilities there are? Is it just $\displaystyle \frac{possibilities}{52\times 51\times 50\times 49\times 48}$ ?