24 feet is a long span....almost too long for dimension lumber, presuming your pitch will be 2 or 3 over 12. And I also presume that this is being built in the deep South, where there is NO possiblility of a snow load.
My load charts only go up to a 6X12 clear fir beam (species group A) on 4 foot centers, which shows a maximum of a 16 foot span.
Personally, I would not try to clear span this distance without going to a glu-lam, an engineered truss or a composit engineered beam. You also have the concern of your ledger, which you should probably not secure to your house due to weight (i.e., it should be supported independent of the house), as well as the the header at the other end. And then there is the issue of wind lift on such a large area..........Whew!
For this much span, I would retain a structural engineer and be prepared to spend some pretty hefty money for the kind of structure you will need. Engineering mistakes here could prove to be very costly.
Best wishes and I hope this works for you. That much clear spanned patio will look very impressive.