Permutations & combinations: notes and practice questions

A hotel has five vacant rooms in a row, labelled 1 to 5. Four new guests, Alice, Ben, Chloe, and David, are to be assigned to these rooms. Alice and Ben are a couple who have recently had an argument and wish to be in separate rooms.

(a) The rooms are large suites, and each suite can accommodate all four guests. Find the number of ways the guests can be assigned to the rooms if Alice and Ben must be in different rooms.

(b) Each room can only accommodate one guest. Find the number of ways the guests can be assigned to the rooms if Alice and Ben must not be in adjacent rooms.

A project manager has $n$ employees available for a new project. The employees are to be divided into two teams, Team Alpha and Team Beta. For the project to be successful, Team Alpha must have exactly four members and Team Beta must have at least four members.

The manager will randomly assign four employees to Team Alpha, with the rest forming Team Beta.

Write down an expression for the number of ways that the teams could be formed.

Two of the employees, Chloe and David, have a history of conflict and cannot be in the same team. The manager agrees to this condition, and finds that this restriction reduces the number of possible team formations to two-fifths of the original number.

Determine the value of $n$.

A school is organizing a science fair. There are 6 display booths available, arranged in a single row. Five different projects are to be displayed: Biology, Chemistry, Physics, Earth Science, and Mathematics.

The booths are large and can hold multiple projects. The Physics and Chemistry projects use sensitive equipment and cannot be placed in the same booth.

(a) Find the number of ways of placing the five projects in the six booths.

(b) Each booth may only contain one project. The Physics and Chemistry projects cannot be placed in adjacent booths. Find the number of ways of placing the five projects in the six booths.

A new automated manufacturing process produces components. The time, in minutes, taken for a critical assembly step is modelled by a continuous random variable $X$, with a probability density function defined by

$f(x) = \begin{cases} \frac{2}{\pi \sqrt{9-x^2}} & 0 \leq x \leq 3 \\ 0, & \text{otherwise.} \end{cases}$

Find the exact value of $E(X)$.

Find $P(X < 1.5)$.

The assembly step is considered "efficient" if it takes less than 1.5 minutes. Each assembly step is independent. Determine the least number of assembly steps required to be 99% sure of at least one efficient step.

Ten assembly steps were conducted.

Find the probability that exactly three steps were efficient.

Write down the number of ways these three efficient steps could have occurred consecutively in a batch of 10.

Now consider a batch of $n$ assembly steps where it is given that exactly three efficient steps have occurred.

Write down an expression for the number of ways these three efficient steps could have occurred consecutively.

Find the greatest value of $n$ such that the probability of three consecutive efficient steps is more than 0.05, given that exactly three efficient steps have occurred in the batch.

Seven students are lining up for a class photograph. Two specific students, Ben and Chloe, insist on standing next to each other, with Ben always to Chloe's immediate left.

Find the total number of possible ways the seven students can line up under this condition.

In a different arrangement of the same seven students, two other students, Daniel and Emily, have a preference. Daniel must stand somewhere to the left of Emily (not necessarily immediately to the left).

Find the total number of possible ways the seven students can line up under this condition.

A student is exploring some properties of combinations.

(a) Show that $^nC_r = \frac{n}{r} \, ^{n-1}C_{r-1}$ for integers $n \ge r \ge 1$.

(b) Show that $^nC_r \times ^rC_m = ^nC_m \times ^{n-m}C_{r-m}$ for integers $n \ge r \ge m \ge 0$.

A museum curator is arranging 5 unique ancient scrolls in a row of 10 empty display cases. For a special exhibit, these 5 scrolls must be displayed next to each other in a continuous block.

Find the number of ways these 5 scrolls can be arranged in this row.

For a different display, the curator returns the scrolls to the same row of 10 empty display cases. One of the scrolls, "The Cursed Amulet", must not be displayed next to any of the other four scrolls. The other four scrolls do not have to be displayed next to each other.

Find the number of ways these 5 scrolls can now be arranged in this row.

A group of 8 students are participating in a science fair. They are to be arranged in a single row for a photograph. Find the number of different arrangements possible.

A company is creating a new system for employee access codes. Each code must be between 4 and 6 characters long, inclusive. The characters are chosen from the set {A, B, C, 1, 2, 3, 4}, without repetition. Each code must start with a letter and end with an even number. Determine the total number of different access codes that can be created.

A school is forming a student council committee of 6 members. The candidates consist of 8 students from Grade 12 and 6 students from Grade 11. The committee must have at least one student from Grade 11, and there must be more Grade 12 students than Grade 11 students. Calculate the number of different committees that can be formed.

A company has a server rack with 7 servers arranged in a single row. Five distinct software applications, App A, App B, App C, App D, and App E, are to be deployed. Each server is powerful enough to run all five applications. However, App A and App B are incompatible and cannot be deployed on the same server.

(a) Find the number of ways of deploying the applications.

(b) In a different scenario, six distinct applications, App 1 to App 6, are to be deployed. Each server can only run one application. App 1 and App 2 require significant network bandwidth and must not be deployed on adjacent servers. Find the number of ways of deploying these six applications.

A software development team consists of four senior engineers and six junior technicians.

The team members are to be placed in a line to have their photograph taken.

(a) Calculate the number of ways the team members can be placed if

(i) there are no restrictions;

(ii) all senior engineers must be placed next to each other.

Five members of the team are selected to work on a critical project. Find the number of possible selections that contain at least three senior engineers.

A bag contains five tiles, each with a different number from the set $\{1, 2, 3, 4, 5\}$. Two tiles are drawn from the bag at random, one after the other, without replacement. The outcome is recorded as an ordered pair.

(a) State the number of possible outcomes.

(b) Find the probability that the number on the second tile is greater than the number on the first tile.

(c) Find the probability that the sum of the numbers on the two tiles is an even number.

(a) A university library contains a collection of $25$ new fiction books, of which $6$ are fantasy novels. A student randomly selects $7$ books from this collection to take home.

Find the probability that the selection will contain no fantasy novels.

(b) Find the probability that the selection will contain at least two fantasy novels.

A local cinema has a row of $10$ available seats for a special movie screening. Four friends, Alex, Ben, Chloe, and David, want to sit in this row.

Find the number of possible ways that they can be seated in this row if they decide not to sit together as a group of four.

The next day, Alex, Ben, Chloe, and David are joined by two additional friends, Emily and Frank, for another screening in the same cinema row of $10$ seats. This time, Alex, Ben, Chloe, and David decide to sit next to each other as a group of four.

Find the number of possible ways that these $6$ people can be seated.