However once in the air, they had to move as the crew were preparing food and drinks.
Paula Taylor told the show that she, her husband and 10-year-old daughter had got to the airport early, in June, to make sure they were seated together.
They searched for their seats but to no avail. "We couldn't believe it when we realised that there genuinely weren't any seats for us", Mrs Taylor said.
A free seat was found for Brooke after all passengers had boarded while Mr and Mrs Taylor were given flip-up seats in the crew section.
They were soon joined by Brooke and even the plane's co-pilot, who Paula said thanked the family for their "cooperation and understanding" in hard conditions.
But despite the efforts of staff to minimize their discomfort, the family still spent the flight on the floor.
The episode of Rip-Off Britain: Holidays will be broadcast on BBC One on Tuesday at 9.15am. "It's just not an experience I ever want to repeat".
The family, from Alcester, Warwickshire, had paid £1,300 and were flying from Mahon in Menorca to Birmingham with TUI airlines.
TUI said it was "sorry for the way the situation was initially handled" and will contact the family directly to apologise.
"Unfortunately a last minute aircraft change meant that the seats the family was originally assigned were unavailable as the alternative aircraft had a different seating configuration".
But Paula said the company was unhelpful when she first complained, even after she sent them photographic proof of what had happened.
Later, TUI offered them £30 as a "goodwill gesture", while the Taylors responded by offering having their tickets refunded.
Under Civil Aviation Authority rules, flyers are not permitted to be left unseated at any stage of a flight.
They turned down the option of a later flight and opted for the row with missing seats.