Here’s the breakdown of the problem and how to solve each part:

Given:

• Anita invites 10 friends.
• There are a couple, 4 men, and 4 women.
• The total number of people is 11 (Anita + 10 friends).
• They are seated around a circular dinner table.

General Circular Permutation Formula:

For $n$ people seated around a circular table, the number of ways to arrange them is: $(n-1)!$ For 11 people, the general arrangement is: $(11-1)! = 10!$

(a) Men insist to sit next to each other

• There are 5 men, treating them as a single unit.
• This unit + the remaining 6 people form 7 groups.
• The 7 groups can be arranged in: $(7-1)! = 6!$
• The men within their unit can be arranged among themselves in: $5!$
• Total ways: $6! \times 5!$

(b) The couple insists to sit next to each other

• The couple is treated as one unit.
• This unit + the remaining 9 people form 10 groups.
• The 10 groups can be arranged in: $(10-1)! = 9!$
• The couple can swap seats within their unit in: $2!$
• Total ways: $9! \times 2!$

(c) Men and women sit alternately

• There are 5 men and 5 women, plus Anita.
• We fix Anita’s position first.
• The 5 men can be arranged among themselves in: $5!$
• The 5 women can be arranged among themselves in: $5!$
• Total ways: $5! \times 5!$

(d) Photo-shoot with Anita and her husband together

• Since this is a row arrangement, the number of ways to arrange all 11 people in a row is: $11!$
• Treating Anita and her husband as one unit, we get 10 groups.
• The 10 groups can be arranged in: $10!$
• The couple can switch places within their unit in: $2!$
• Total ways: $10! \times 2!$

Would you like a detailed step-by-step breakdown for any part? 🤓

Related Questions:

1. How many ways can the couple sit opposite each other at the table?
2. If three specific friends must sit next to each other, how does the calculation change?
3. How many ways can only the women sit together at the table?
4. If everyone is seated randomly, what is the probability that the couple sits together?
5. How many ways can they be arranged in a row if the men and women must be in separate groups?

Tip:

When solving circular permutation problems, remember that fixing one person (like Anita) simplifies calculations! 🎯