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You will get better prices and also help and advice if you book by phone through this or another specialist cruise agent. Cruises are not like ordinary package hols where you get an online booking discount. (There are indeed cabins which sleep 3 or 4, but you may find they are already fully booked for the August school holiday period.).
All ships have cabins which accomodate 3 or 4 persons. Cabins which take more than 2 occupancy are usually marked on the deck plans with signs such as star, triangle or cross, etc which indicates whether or not they take 3 or 4. The legend next to the deck plan should explain whether or not this consists of upper pullmans or sofa beds. You will usually find that insides are nearly always upper bunks for the 3rd/4th passenger, whilst outsides can be either. On most ships the Balcony cabins do not have upper bunks and usually consist of a sofa bed. Now personally I have mixed feelings about a Balcony cabin with a sofa bed. Speaking as someone who has an 18 year old - You have a lovelyBalcony with a seating area, but when the Sofa Bed has been made up early evening you lose all your floor space - not much room for three people to get dressed. Also if your 17 year old (like my 18 year old!) is up until the early morning and sleeps in until lunch, you are still clambering around the sofa bed. I prefer the bunk option myself as then my sons can sleep all they like suspended up in the air without affecting the rest of us. Hope this is of some help to you. Someone else may have a different view on this.
Now I have just gone back to look at this. Not sure which ship you are actually looking at but on Oceana, I did find some Balcony Staterooms: Staterooms A640, A641, A730, A732, A735, A737, B730, B732, B735, B737 consist of a fixed lower twin plus one lower and two upper pullman berths which fold away. These staterooms are intended for 3/4 berth use and are fixed single berths. So there are some balcony cabins whihc do have upper pullmans but it does state that you cannot then have your twins made upinto a double. Hope I havent confused you even more. Really you need to speak to the people at Cruise Co by phoning the number at the top of this page. They really are very knowledgeable and helpful.