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:

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

and 3 from the other rank:

So there are:

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 ?