water coming in from deck joists

Had a couple days of steady rain and noticed water coming in from deck joists. You can see four 2x10s bolted together inside the house. Together they extend outside the house about two feet. then just two of them continue to the edge of the deck. So water must be getting between them and dripping down into the house. I will find a temporary way to keep the water further away from the house for now. The question is when i have the deck replaced next year, do I allow a contractor to use this method to support the new deck? My last house had a deck that was attached to the side of the house by a ledger board, then deck joists with hangers. So, i'm not familiar with joists running inside the house and bolted to interior floor joists. It seems using flashing would be easier with a ledger board. Not sure how you would use flashing with joists running into the house.