# Combinatorics problem

1. Oct 24, 2016

### Mr Davis 97

1. The problem statement, all variables and given/known data
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?

2. Relevant equations

3. 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?

2. Oct 24, 2016

### QuantumQuest

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?

3. Oct 24, 2016

### haruspex

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