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

[Mr Davis 97]

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?

 

[QuantumQuest]

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?

 

[haruspex]

we subtract from this 3!⋅6!,

Please explain your reasoning for that number. Remember, you have already combined the twins into one entity.