Arrangements with couples/pairs

6 Jai and his wife Kaz are having a party. Jai has invited five friends and each friend will bring his wife.

1. At the beginning of the party, the 12 people will stand in a line for a photograph.
   1. How many different arrangements are there of the 12 people if Jai stands next to Kaz and each friend stands next to his own wife?
   2. How many different arrangements are there of the 12 people if Jai and Kaz occupy the two middle positions in the line, with Jai's five friends on one side and the five wives of the friends on the other side?
2. For a competition during the party, the 12 people are divided at random into a group of 5, a group of 4 and a group of 3 . Find the probability that Jai and Kaz are in the same group as each other.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7

1. Seven friends together with their respective partners all meet up for a meal. To commemorate the occasion they arrange for a photograph to be taken of all 14 of them standing in a line.
   1. How many different arrangements are there if each friend is standing next to his or her partner?
   2. How many different arrangements are there if the 7 friends all stand together and the 7 partners all stand together?
2. A group of 9 people consists of 2 boys, 3 girls and 4 adults. In how many ways can a team of 4 be chosen if
   1. both boys are in the team,
   2. the adults are either all in the team or all not in the team,
   3. at least 2 girls are in the team?

3 Mr and Mrs Keene and their 5 children all go to watch a football match, together with their friends Mr and Mrs Uzuma and their 2 children. Find the number of ways in which all 11 people can line up at the entrance in each of the following cases.

1. Mr Keene stands at one end of the line and Mr Uzuma stands at the other end.
2. The 5 Keene children all stand together and the Uzuma children both stand together.