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On Chinese New Year celebration, a big family of 18 members meet at an exclusive hotel restaurant to have their special lunch. They have made exclusive reservations and the hotel restaurants have already prepared for them one round table which can consist ten of them and a square table which can consist eight of them. The family members consist of Mr. and Mrs. Lim, both of Mrs. Lim’s parents, Mr. Lim’s two brothers and four sisters, Mrs. Lim’s two adult nephews and lastly Mr. Lim’s four daughters and two sons all of whom aged 15 and above. Considering the placements of the people at their respective tables only (i.e. neglect where two people are from each other if they are from separate tables), determine how many ways the family can be seated if Mr. and Mrs. Lim must sit in between Mrs. Lim’s parents, and each group having the same relationship with either Mr. or Mrs. Lim (as given in the scenario) must sit at the same table; e.g. Mrs. Chong’s two adult nephews must sit at the same table. Remarks: Considering only the positioning of the people with respect to each other for the square table the same way we do for the round table (i.e. neglect the sides of the square table).

Answer :

There are 1,036,800 ways the family can be seated according to the given conditions.

There are two tables available for the family to be seated: a round table that can accommodate ten people and a square table that can accommodate eight people. The family consists of 18 members.

To determine the number of ways the family can be seated, we need to consider the given conditions:
1. Mr. and Mrs. Lim must sit in between Mrs. Lim's parents.
2. Each group that has the same relationship with either Mr. or Mrs. Lim must sit at the same table.

Let's break down the seating arrangement step-by-step:

Step 1: Seating arrangement for Mrs. Lim's parents
Since Mr. and Mrs. Lim must sit in between Mrs. Lim's parents, we have the following arrangement:

Mrs. Lim's father - Mrs. Lim's mother - Mr. Lim - Mrs. Lim

Step 2: Seating arrangement for Mr. Lim's siblings
Mr. Lim has two brothers and four sisters. They all have the same relationship with Mr. Lim. Since the round table can accommodate ten people, we can seat Mr. Lim and his siblings together at this table. There are 6 siblings, including Mr. Lim, so we have the following arrangement:

Mr. Lim - Mr. Lim's siblings

Step 3: Seating arrangement for Mrs. Lim's nephews
Mrs. Lim has two adult nephews. According to the given condition, they must sit at the same table. Since the round table is already occupied by Mr. Lim and his siblings, we can seat Mrs. Lim's nephews at the square table. There are two seats available, so we have the following arrangement:

Mrs. Lim's nephews

Step 4: Seating arrangement for Mr. Lim's children
Mr. Lim has four daughters and two sons, all aged 15 and above. They all have the same relationship with Mr. Lim. Since the round table is already occupied, we can seat Mr. Lim's children at the square table. There are six seats available, so we have the following arrangement:

Mr. Lim's children

Now, let's calculate the number of ways to arrange each group at their respective tables:

For the round table: There are 6 seats available for Mr. Lim and his siblings. The number of ways to arrange them is 6!.

For the square table: There are 2 seats available for Mrs. Lim's nephews and 6 seats available for Mr. Lim's children. The number of ways to arrange them is 2! * 6!.

To determine the total number of ways the family can be seated, we multiply the number of ways for each table:

Total number of ways = Number of ways for the round table * Number of ways for the square table

Total number of ways = 6! * 2! * 6!

Simplifying this expression, we get:

Total number of ways = 720 * 2 * 720

Total number of ways = 1,036,800

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