Use the notation (𝑛 𝑟) and 𝐶𝑟 𝑛 to represent the number of ways of selecting 𝑟 objects from 𝑛 distinct objects where order is not important. 𝐶𝑟 𝑛 = (𝑛 𝑟) = 𝑛! 𝑟!(𝑛−𝑟)!

A research team is selecting participants for three simultaneous clinical trials: Trial A requires 4 participants, Trial B requires 5 participants, and Trial C requires 3 participants. The table below shows the number of eligible candidates from different age groups who have volunteered. The team must select all participants for Trial A from the 18–35 age group only, all participants for Trial B from the 36–50 age group only, and all participants for Trial C from the 51–65 age group only. No person can participate in more than one trial. (a) Determine the total number of different ways the research team can select participants for all three trials simultaneously. Express your answer using combination notation, then evaluate. (2 marks) (b) After the initial selection, one participant from Trial A withdraws. The team decides to replace this participant by selecting from the remaining candidates in the 18–35 age group. Show that the number of ways to complete the replacement is $\binom{8}{1}$. (1 mark) (c) Given that the replacement has been made, the research team now wishes to form a review committee of 6 people chosen from all participants across the three trials (now totaling 12 participants). Determine the probability that the review committee contains exactly 2 participants from Trial A, exactly 3 from Trial B, and exactly 1 from Trial C. (2 marks)

Use the formula $C_r^n = \frac{n!}{r!(n-r)!}$ to determine the value of $C_5^{12}$.

A regular octagon has 8 distinct vertices labelled A, B, C, D, E, F, G, H. Which expression represents the number of different ways of selecting any 4 vertices from these 8 vertices?

A regular hexagon has 6 distinct vertices labelled A, B, C, D, E, F. Use the notation $\binom{n}{r}$ to express the number of different ways of selecting 3 vertices from the 6 available vertices.

A committee is selecting 4 members from a group of 8 distinct candidates labelled A, B, C, D, E, F, G, and H. The diagram shows one possible selection of 4 members (shaded vertices). How many different ways can 4 members be selected from the 8 candidates? Express your answer using the notation $\binom{n}{r}$.

A regular octagon has 8 vertices labelled A, B, C, D, E, F, G, H. Four of these vertices need to be selected to form a quadrilateral. Which expression using the notation $\binom{n}{r}$ correctly represents the total number of different quadrilaterals that can be formed?

A committee is to be formed by selecting 4 members from a group of 8 people. Use the notation $\binom{n}{r}$ to express the number of different committees that can be formed.

A committee of 12 people must select a subcommittee consisting of a chairperson, a secretary, and 3 additional members. a) Use combination notation to express the number of ways of selecting the 3 additional members from the 10 remaining people (after the chairperson and secretary have been chosen). (1 mark) b) Use the result from part (a) to determine the total number of ways the subcommittee can be formed if the chairperson and secretary are chosen first from the 12 people. Show the values substituted into your expression. (2 marks)

A regular octagon has 8 distinct vertices labelled A, B, C, D, E, F, G, H. Express the number of ways of selecting any 4 vertices from the 8 vertices using the notation $\binom{n}{r}$.

A netball coach must select a starting team of 7 players from a squad of 12 available players for the grand final match. Three of the players in the squad—Aisha, Bella, and Chloe—are the only goal shooters available. a) Calculate the total number of different starting teams that can be selected from the squad of 12 players. (1 mark) b) Calculate the number of different starting teams that include no goal shooters. (1 mark) c) Use the results from parts (a) and (b) to determine how many different starting teams can be formed if at least one goal shooter must be included in the team. Use appropriate notation to express your calculation method. (1 mark)

A school committee must select 5 representatives from a group of 12 students to attend a state conference. After the initial selection is made, it is discovered that exactly 3 of the selected representatives must be from the 7 senior students in the group, with the remaining 2 representatives chosen from the 5 junior students. Determine the total number of ways the committee can form this selection.