N2017 P2 Q6

A children’s game is played with \(20\) cards, consisting of \(5\) sets of \(4\) cards. Each set consists of a father, mother, daughter and son from the same family. The family names are Red, Blue, Green, Yellow and Orange. So, for example, the Red family cards are father Red, mother Red, daughter Red and son Red.

The \(20\) cards are arranged in a row.

1. In how many different ways can the \(20\) cards be arranged so that the \(4\) cards in each family set are next to each other?[2]
2. In how many different ways can the cards be arranged so that all five father cards are next to each other, all four Red family cards are next to each other and all four Blue family cards are next to each other?[3]

The cards are now arranged at random in a circle.

3. Find the probability that no two father cards are next to each other.[4]