Solve problems that involve combinations.

A school debate team consists of 15 students: 9 girls and 6 boys. A team of 5 students must be selected to represent the school at a regional competition. The selection rules require that the team must include at least 2 boys and at least 2 girls. Determine the number of different teams that can be selected.

A school committee is selecting members from a group of students to form a subcommittee. The diagram below shows the available students divided into three year levels. The committee must select exactly 3 students from Year 10 and exactly 2 students from Year 11 to form a 5-member subcommittee. Calculate the total number of different subcommittees that can be formed.

A restaurant offers a tasting menu where customers select 3 appetizers from 8 available, 4 main courses from 12 available, and 2 desserts from 6 available. Calculate the total number of different tasting menus that can be created.

A student council consists of 15 members. A delegation must be formed to attend a regional conference. (a) Determine the number of different ways a delegation of 6 members can be selected from the 15 members. (1 mark) (b) The delegation must include at least 2 of the 4 executive officers (president, vice-president, treasurer, and secretary). Determine how many different delegations can be formed under this condition. (2 marks)

A school committee must select exactly 3 students from a group of 8 Year 10 students and exactly 2 students from a group of 6 Year 11 students to form a 5-member subcommittee. How many ways can the 3 Year 10 students be selected?

A committee of 5 people is to be selected from a group of 7 teachers and 4 students. The committee must include at least 3 teachers and at least 1 student. How many different committees can be formed?