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

In summary, the Jones family has 5 boys and 3 girls, with 2 of the girls being twins. They need to be seated in a row of 8 chairs, with the twins insisting on sitting together and their other sister refusing to sit next to either of them. Using a "count the complement" technique, there are ##2!\times 7!## ways to arrange the twins together. However, to account for the third sister not being adjacent, we need to treat the three sisters as one entity. Therefore, there are 3! ways to arrange the three sisters together and 6! ways to arrange the boys and the three sisters together. Hence, the total number of arrangements is ##2!\times 7
  • #1
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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  • #2
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?
 
  • #3
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.
 

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

1. What is Combinatorics?

Combinatorics is a branch of mathematics that studies the different ways in which objects can be combined, arranged, or selected. It deals with counting and organizing objects and understanding their properties.

2. What are some common types of Combinatorics problems?

Some common types of Combinatorics problems include permutations, combinations, and the binomial theorem. Other types include graph theory, enumeration, and optimization problems.

3. How is Combinatorics used in real-world applications?

Combinatorics has many practical applications in fields such as computer science, engineering, economics, and biology. It is used to solve problems involving optimization, scheduling, coding, and data analysis, among others.

4. What are some useful techniques for solving Combinatorics problems?

Some useful techniques for solving Combinatorics problems include the use of counting principles such as the multiplication and addition rules, as well as various formulas and theorems such as the binomial theorem and the inclusion-exclusion principle.

5. How can I improve my skills in solving Combinatorics problems?

Practice is key to improving your skills in solving Combinatorics problems. Additionally, studying and understanding the fundamental principles and techniques of Combinatorics, as well as working through a variety of problems and seeking help when needed, can also help improve your skills.

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