Exercises: Introduction

A restaurant offers a set menu with $4$ starters, $3$ main courses and $5$ desserts. How many different ways can a full three-course meal consisting of a starter, a main course and a dessert be ordered?

Draw the tree diagram for the experiment consisting of tossing a fair coin and then rolling a tetrahedral die (with its faces numbered from $1$ to $4$). How many possible outcomes does this compound experiment have?

You can do it on paper and send a photo with your answer.

In a classroom library there are $12$ science fiction novels, $8$ historical novels and $5$ biographies. If you only have time to read one book this weekend, how many different options do you have to choose from?

Calculate the value of the following numerical expressions using the properties of factorials:

• $\frac{8!}{6!}$

• $4! + 3!$

A bicycle brand manufactures a model that is available in $3$ different frame sizes, $5$ colours and with $2$ different types of brakes. How many different versions of this bicycle model are on the market?

To go from city A to city B there are $3$ different roads, and to go from city B to city C there are $2$ different roads.

• In how many ways can you travel from A to C, ensuring you pass through B?

• Draw the tree diagram corresponding to this journey.

You can do it on paper and send a photo with your answer.

In a chest of drawers, there are loose socks of $3$ different colours (black, white, and blue). If there is a power outage and you take out socks in the dark, what is the minimum number of socks you must pull out to guarantee having at least one pair of the same colour?

Express the following products using factorial notation:

• $15 \times 14 \times 13$

• $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$

Cian is solving a simplification problem involving factorials and writes the following working on the board:

1. a) Pinpoint the algebraic error Cian made and explain why it is not valid to simplify the fraction in that way.
2. b) Solve the expression correctly by applying the properties of factorials (expressing $7!$ in terms of $5!$ or factoring out the common factor) and find its final numerical value.

To protect a bicycle at high school, a combination lock with $3$ rotating wheels is used, each with the digits from $0$ to $9$. The owner states that the code cannot start with $0$ and that the three digits must be distinct from each other.

1. a) Before performing any exact calculations, intuitively estimate whether the total number of possible codes will be less than $300$, between $300$ and $700$, or greater than $700$. Briefly justify what your initial intuition is based on.
2. b) Model the situation using the multiplication principle and calculate the exact number of different codes that satisfy these conditions.
3. c) Check whether your initial estimate was correct and reflect on how the result changes if digits were allowed to be repeated (keeping the first digit different from $0$).

A password for a computer system must consist of $2$ letters of the alphabet (out of $26$ possible letters) followed by $3$ digits (from $0$ to $9$), or $3$ letters followed by $2$ digits. If repetition of both letters and digits is allowed, how many different passwords can be created in total?

Two basketball teams, Alfa and Beta, play a best-of-$3$ playoff (the first team to win $2$ games qualifies). Draws are not possible. Draw the tree diagram with all possible outcomes of the playoff. In how many different ways can the series of games unfold?

You can do it on paper and send a photo with your answer.

Simplify the algebraic expression $\frac{(x+2)!}{x!}$ and then solve the equation $\frac{(x+2)!}{x!} = 72$.

Using only the digits from the set $\{1, 2, 3, 4, 5\}$:

• How many numbers with $4$ distinct digits can be formed?

• How many of these formed numbers are even?

In a secondary school, there are $400$ students enrolled.

• Justify mathematically why you can guarantee that at least two students share the same birthday.

• What is the minimum number of students needed at the school to guarantee that at least three of them share the same birthday? (Assume non-leap years of $365$ days).

A multiple-choice exam consists of $10$ questions. Each question has $4$ answer options, only one of which is correct.

• In how many different ways can the exam be answered if selecting an option for every question is mandatory?

• In how many ways could it be answered if leaving questions blank is allowed?

An urn contains $3$ red balls and $2$ blue balls. Two balls are drawn sequentially without replacement. Draw the tree diagram for the experiment and calculate how many distinct outcomes can be obtained if the order of extraction and the colour of the balls are taken into account.

You can do it on paper and send a photo with your answer.

Look at the grid in the figure. A robot moves from point A to point B, moving only to the right or upwards along the grid lines. How many different paths can the robot take to reach its destination?

In a youth book club, they want to assign each member a card with a personalized code. The code is always composed of an uppercase vowel (chosen from the set $\{A, E, I, O, U\}$) followed by a sequence of $k$ distinct odd digits (chosen from the set $\{1, 3, 5, 7, 9\}$).

The person in charge of the database notices that, under the set conditions, exactly $300$ different codes can be generated.

1. a) State a general expression in terms of $k$ for the total number of possible codes by applying the multiplication principle.
2. b) Solve the inverse problem: determine, with reasoned arguments, how many odd digits ($k$) each code must contain to obtain exactly those $300$ codes.

In a secondary school in Cork, there are $750$ students enrolled in total. To manage the laptop loan system, an ID tag is generated for each student consisting of a letter from an alphabet of $27$ letters followed by an even digit belonging to the set $\{0, 2, 4, 6, 8\}$.

1. a) Model the situation using the multiplication principle to calculate how many different ID tags can be issued with this format.
2. b) Use the pigeonhole principle to rigorously justify why it is certain that there will be students assigned identical tags.
3. c) Determine the minimum number of students who, with absolute mathematical certainty, will share the exact same tag in that school.

A flag with $3$ vertical stripes is to be designed using a palette of $5$ available colours.

• In how many ways can it be coloured if two adjacent stripes cannot have the same colour?

• In how many ways can it be coloured if, in addition to the previous condition, the first and last stripes must also be of different colours?

Solve the following equation involving factorials, knowing that $n$ is a natural number:

$5$ points are chosen at random inside a square of side length $2\text{ cm}$. Prove, using the pigeonhole principle, that there will always be at least two points whose distance apart is less than or equal to $\sqrt{2}\text{ cm}$. (Hint: divide the square into strategic regions).

In the decimal number system, consider all integers from $1000$ to $9999$ (inclusive). How many of these numbers contain the digit $7$ at least once in their representation?

Two friends, Ionut and Diana, play a board game consisting of several rounds. The overall winner of the game will be the first person to win $2$ consecutive rounds or $3$ rounds in total across the match. Knowing that draws are not possible in any round, draw the corresponding tree diagram and determine the maximum number of rounds they can play and how many different game paths there are.

You can do it on paper and send a photo with your answer.

In a class of $20$ students, a leadership team consisting of a president, a vice-president, and a secretary must be chosen (distinct positions that cannot be held by the same person simultaneously). Once the leadership team has been chosen, a sports representative must be selected from the remaining $17$ students. In how many different ways can this entire selection process be carried out?

A palindrome number is a number that reads the same forwards and backwards (for example, the number $35153$).

• How many $5$-digit palindrome numbers are there in total?

• How many of these $5$-digit palindrome numbers are even numbers?

Prove algebraically the following identity, valid for any natural number $n \ge 1$:

Once proven, apply this result to simplify and calculate the exact value of the expression $\frac{10! + 9!}{11!}$.

In the emergency department of a local hospital, the weekend on-call shifts must be covered by assigning one doctor for Saturday and a different one for Sunday, chosen from a team of $6$ available professionals: Aoife, Seán, Niamh, Cian, Saoirse, and Oisín. Due to rest requirements after a previous procedure, Aoife and Seán cannot overlap on the same weekend (that is, it cannot happen that one is on duty on Saturday and the other on Sunday).

To calculate the number of possible valid shifts, two residents suggest different strategies:

• Strategy A (Direct counting by cases): Add the options in which neither of the two incompatible doctors participates and the options in which exactly one of them participates.

• Strategy B (Complementary counting): Calculate all possible assignments of distinct doctors for Saturday and Sunday without restrictions and subtract the forbidden cases where Aoife and Seán overlap.

1. a) Work out the step-by-step calculations in detail following Strategy A and then following Strategy B, verifying that both methods lead to the same numerical result.
2. b) Compare both procedures and justify which of the two is more advantageous and less prone to errors if the team consisted of $25$ doctors instead of $6$, maintaining the same restriction between Ana and Blas.

Met Éireann, the Irish Meteorological Service, classifies wind observations at a station in the Wicklow Mountains according to two simultaneous variables: direction, which takes one of the $8$ main compass points (N, NE, E, SE, S, SW, W, NW), and strength, categorized into $3$ levels (light, moderate, strong).

1. a) Every day, three independent measurements are recorded: at 08:00 h, 14:00 h, and 20:00 h. A "daily report" consists of the ordered triple of these three full measurements (direction and strength in each). Apply the multiplication principle to calculate how many different daily reports can theoretically be recorded.
2. b) A researcher states: "If we analyze only the 14:00 h measurement throughout a leap year ($366$ days), it is mathematically certain that there will be at least one combination of direction and strength that will have repeated a minimum of $16$ times". Formally justify whether this statement is true or false using the pigeonhole principle (clearly identifying which are the objects or pigeons and which are the groups or pigeonholes).
3. c) What is the minimum number of days for which the 14:00 h measurements should be recorded to guarantee with absolute mathematical certainty that some combination has repeated at least $20$ times? Justify your answer.