The Jones family has 5 boys and 3 girls, and 2 of the girls are twins. In how many ways can they be seated in a row of 8 chairs if the twins insist on sitting together, and their other sister refuses to sit next to either of her sisters?
The Attempt at a Solution
I thought that I could use a "count the complement" technique. First, we would count the the number of ways to just have the two twins paired together. This would be ##2! \cdot 7!## ways. However, this over counts because it includes the pairs where the other sister is adjacent. Thus, we subtract from this ##3! \cdot 6!##, which is the number of arrangements where the other sister is adjacent to the other sisters. This gives 5760. However, this is not the right answer. What am I doing wrong?