Combinatorics Problem: Seating a Family of 8 with Twins and Siblings

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Mr Davis 97
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Homework Statement


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?

Homework Equations

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?
 
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Mr Davis 97 said:

Homework Statement


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?

Homework Equations

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!⋅7!2!⋅7!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!⋅6!3!⋅6!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?

You are right about the number of ways to just have the two twins paired together (##2!\times 7!##). But then, to account for the third sister not being adjacent, you have to think more carefully. As a hint, I recommend to treat the three sisters together. Now, how many ways are there to arrange this with the boys? How many about the three sisters together?
 
Mr Davis 97 said:
we subtract from this 3!⋅6!,
Please explain your reasoning for that number. Remember, you have already combined the twins into one entity.