Solve problems that involve permutations.

A theatre company is staging a play with seven principal actors. The director requires that actors P, Q and R must not stand adjacent to one another in the final curtain call line-up. (a) Determine the total number of possible arrangements of all seven actors without restriction. (1 mark) (b) Determine the number of arrangements in which actors P, Q and R all stand adjacent to one another. (1 mark) (c) Hence, determine the number of arrangements that satisfy the director's requirement. (1 mark)

A theatre company is planning seating arrangements for a special performance. The table below shows the number of VIP guests from different organisations who must be seated in a single row of 12 chairs. Determine the number of distinct seating arrangements possible if all guests from the same organisation must sit together as a group.

A school committee must arrange 7 students in a row for a photograph. Three of the students, Alex, Bailey, and Casey, must stand together as a group. a) Determine the number of ways the 7 students can be arranged in the row, given this constraint. (2 marks) b) If Alex must stand in the middle position of the three students who stand together, determine how many different arrangements are now possible. (1 mark)

How many different four-letter arrangements can be formed using the letters shown in the diagram, if each letter may be used at most once?

A library shelf has 8 different mathematics textbooks that must be arranged in a row. Three of these textbooks are statistics books that must be kept together as a group, while the remaining 5 textbooks can be arranged in any order. Calculate the number of different arrangements possible.