2021 RI J2 CT Q9

A card game called ‘Happy Family’ is played with \(28\) cards, consisting of \(7\) sets of \(4\) cards. Each set consists of a father, a mother, a son and a daughter from the same family. The family names are Painter, Postman, Plumber, Butcher, Carpenter, Singer and Teacher. So for example, the complete set of Teacher family cards consist of father Teacher, mother Teacher, son Teacher and daughter Teacher.

The objective of the game is to collect as many complete sets of family cards as possible. At the end of the game, all the cards are collected. Each player will either be left with no cards or complete sets of family cards. The winner is the one with the most number of complete sets of family cards.

\(A\) and \(B\) play a game of ‘Happy Family’.

1. At the end of the game, \(A\) has exactly \(3\) complete sets of family cards and one of the sets is the Teacher family cards. How many possible combinations of complete sets of family cards can \(B\) have?[1]
2. If \(A\) is the winner at the end of the game, how many possible combinations of complete sets of family cards can \(A\) have?[2]
3. Given that \(A\) has exactly \(3\) particular complete sets of family cards at the end of the game, and he arranges these \(12\)cards in a circle. How many different ways can these cards be arranged so that no two mothers are next to each other?[3]