Understand the notion of a combination as an unordered set of 𝑟 objects taken from a set of 𝑛 distinct objects.

A committee of 4 members is to be selected from a group of 9 volunteers. Which of the following expressions correctly represents the number of different committees that can be formed?

A school debating team has 12 students available for selection. The coach needs to choose 4 students to form a team for an upcoming competition. (a) Explain why the number of ways to select the 4-student team should be calculated using combinations rather than permutations. (1 mark) (b) Determine the number of different 4-student teams that can be selected from the 12 available students. (1 mark) (c) After reviewing the schedule, the coach discovers that 2 of the 12 students cannot attend the competition date. Determine the number of valid 4-student teams that can now be selected. (1 mark) (d) Using your answers from parts (b) and (c), determine the probability that a randomly selected 4-student team from the original 12 students includes at least one student who cannot attend. Express your answer as a fully simplified fraction. (2 marks)

A committee of 5 people is to be formed from a group of 12 people consisting of 7 women and 5 men. The committee must include at least 2 men. (a) Determine the total number of different committees that can be formed without any restrictions. (1 mark) (b) Determine the number of committees that include exactly 1 man. (1 mark) (c) Determine the number of committees that include no men. (1 mark) (d) Hence, determine the number of committees that satisfy the condition of including at least 2 men. (1 mark)

A school librarian is selecting books to create a new display shelf. There are 12 different book titles available, and the librarian needs to choose 4 of them. Calculate the number of different ways the librarian can select 4 books from the 12 available titles.