Specialist Mathematics · Unit 1 · Combinatorics · Permutations (ordered arrangements) and combinations (unordered selections)
AI-marked practice questions tied to QCAA mark schemes for this exact LO. Free to start.
Start free practiceFull questions, answers and worked solutions unlock when you start a free practice session.
A committee of 5 people is to be formed from a group of 8 teachers and 6 students. a) Determine the number of different committees that can be formed if there must be exactly 3 teachers and 2 students. (1 mark) b) Determine the probability that a randomly selected committee contains at least 4 teachers. Express your answer as a fraction in simplest form. (2 marks)
A school committee of 5 members must be formed from a group of 6 teachers and 5 parents. The committee must include at least 3 teachers and at least 1 parent. How many different committees can be formed?
A security company uses 6-digit access codes formed from the digits 1, 2, 3, 4, 5, 6, 7, 8, where each digit can be used at most once in any code. A code is classified as 'secure' if it contains at least two even digits. (a) Determine the total number of distinct 6-digit codes that can be formed. (1 mark) (b) Determine the number of codes that contain exactly zero even digits. (1 mark) (c) Hence determine the probability that a randomly selected code is classified as secure. Express your answer correct to three decimal places. (1 mark)
A quality control team inspects batches of electronic components. Each batch contains 15 components, of which 4 are known to be defective. The team randomly selects 6 components from the batch for detailed testing. (a) Determine the number of ways the team can select exactly 2 defective components in their sample of 6. (b) Use your result from part (a) to determine the probability that exactly 2 defective components are selected, expressing your answer as a fraction in simplest form.
A committee must select a subcommittee from its members. The table below shows the number of members by role and gender. The subcommittee must consist of exactly 3 officers and exactly 2 ordinary members, with at least 1 female member overall. Determine the number of different subcommittees that can be formed.
A school theatre production committee must select a cast of 5 actors from a pool of 8 senior students and 6 junior students. The committee requires that exactly 3 senior students and exactly 2 junior students be chosen. Calculate the number of different possible casts.