Solve problems that involve permutations with restrictions including repeated objects, specific objects grouped together and selection from multiple groups.

A school debating team consists of 8 students: 3 from Year 11 (including twins Alex and Jordan) and 5 from Year 12. (a) Determine the number of ways the 8 students can be arranged in a line for a team photograph if the twins must stand together. (2 marks) (b) Determine the number of ways the 8 students can be arranged in a line if all Year 11 students must stand together as a group. (2 marks)

A school committee of 8 students is to be arranged in a line for a photograph. The committee consists of 3 Year 12 prefects (who must stand together), 2 Year 11 representatives (who must stand at opposite ends), and 3 Year 10 representatives. In how many ways can the 3 Year 12 prefects be arranged within their group?

How many different arrangements are possible?

A student council comprises 12 members: 5 from Year 11 (including 2 prefects) and 7 from Year 12 (including 3 prefects). The council must form a five-member committee to organise a sports carnival. (a) Calculate the number of ways to form the committee if it must include exactly 2 prefects and at least 2 members from Year 12. (2 marks) (b) Calculate the number of ways to form the committee if the 2 Year 11 prefects must both be included or both excluded. (1 mark)

A conference committee is arranging 7 delegates in a row for a photograph. The 7 delegates consist of 3 delegates from Australia, 2 delegates from Canada, and 2 delegates from Japan. (a) Calculate the number of different arrangements if the 2 delegates from Canada must stand next to each other. (1 mark) (b) Calculate the number of different arrangements if all 3 delegates from Australia must stand together. (1 mark) (c) Calculate the number of different arrangements if the 2 delegates from Japan must be separated (not standing next to each other). (1 mark)

A school debating team consists of 12 students: 5 seniors (including the captain and vice-captain) and 7 juniors. (a) Find the number of ways a group of 4 students can be selected from the team if the group must contain exactly 2 seniors and exactly 2 juniors. (1 mark) (b) Find the number of ways the 4 selected students can be arranged in a line for a photograph if the captain and vice-captain must stand next to each other. (1 mark) (c) Hence determine the total number of ways to both select and arrange the group of 4 students if both conditions in parts (a) and (b) apply. (1 mark)