2018 YJC J2 CT1 Q1

\(3\) families went to a cinema to watch a movie together. Each family consists of a father, mother, child and domestic helper. The surnames of the three families are Chua, Lee and Wong. The \(12\) people are seated in a row of \(12\) seats. Find the number of arrangements if

1. the children are seated next to each other,[1]
2. the \(4\) members in each family are seated next to each other,[2]
3. each domestic helper is seated next to the child in the family, and each family is seated together,[2]
4. all three mothers are next to each other, all four members of the Chua family are next to each other and all four members of the Wong family are next to each other.[3]

After the movie, the \(3\) families proceed for a dinner at a round table with \(12\) seats.

5. In how many ways can the \(12\) people sit so that no two children are next to each other?