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

A theatre company is staging a production and needs to select a cast of 8 performers from a pool of 15 available actors. The selection must satisfy the conditions shown in the stimulus. Determine the total number of different ways the cast of 8 performers can be selected.

A theatre company is staging a production and needs to select a cast of 8 performers from a pool of 15 available actors. The selection must satisfy these conditions: • 3 lead roles must be filled from a group of 6 experienced actors • 5 supporting roles must be filled from the remaining 9 actors • Two particular actors, James and Sofia, cannot both be in the supporting cast together. Determine the total number of different ways the cast of 8 performers can be selected.

A committee of $9$ members is to be formed from a group of $12$ teachers and $8$ students. (a) Determine the number of ways to form the committee if exactly $3$ students must be selected and the $3$ students must all be from the same year level, given that $5$ students are Year 11 and $3$ students are Year 12. [2 marks] (b) A different committee of $9$ members is to be formed from the same group, subject to the following restrictions: • At least $5$ teachers must be selected. • If Mr Anderson (one of the teachers) is selected, then Ms Brooks (another teacher) must also be selected. Determine the total number of ways to form this committee. [3 marks]