Exercises: Ways of counting

In a school cafeteria, the set menu allows a choice between 3 starters, 4 main courses and 2 desserts. In how many different ways can a student create a complete lunch consisting of a starter, a main course and a dessert?

A group of 10 friends decides to organise a video game tournament. Prizes will be awarded to the first, second and third place finishers. Given that there can be no ties, in how many different ways can these three trophies be awarded?

To log in to a mobile app, a user must create a 4-digit security code. If the system allows numbers to be repeated and uses the digits from $0$ to $9$, how many different security codes can be generated?

A bookshelf has space to fit exactly 6 textbooks from different Year 2 Bachillerato subjects. In how many different ways can these 6 books be arranged on the shelf if they must all be aligned next to each other?

In a class of 12 students, the form tutor must select 3 of them to be on the student council committee. Given that the three selected students will have the same responsibility, how many different student council committees can be formed?

How many 4-letter words, with or without meaning, can be formed using the letters of the word BALA? (Note that the letter 'A' is repeated).

An ice cream parlour offers 8 different ice cream flavours. If a customer wants to buy a tub with 2 scoops of ice cream of different flavours, how many different combinations can they choose?

A social network requires usernames to always start with a lowercase letter (from the 26 in the English alphabet) followed obligatorily by 2 digits (from $0$ to $9$). How many different usernames can be created under this format?

Marcos wants to calculate how many different words (meaningful or not) can be formed by rearranging all the letters of the word CASAS. On his exam, he writes the following:

"The word has $5$ letters, so there are $P_5 = 5! = 120$ possible arrangements. Since the letter A is repeated $2$ times and the letter S is repeated another $2$ times, to avoid counting duplicate cases I subtract these repetitions: $120 - 2 - 2 = 116$ distinct words."

1. a) Identify the two conceptual errors Marcos makes in his reasoning. In particular, explain in your own words why the correction for repetition should not be made by subtracting, but by dividing.
2. b) State the appropriate combinatorial expression using permutations with repetition, indicate what each term represents, and calculate the actual number of words that can be formed.

An online mobile banking app asks each user to create a $4$-digit numerical access PIN chosen from the numbers from $1$ to $6$, both inclusive.

1. a) Before performing exact calculations, make a qualitative and reasoned estimate: if the banking entity decided to switch from a system that allows repeating digits to one in which the $4$ digits of the PIN must necessarily be distinct, will the number of possible PINs be reduced to more than half, exactly half, or less than half of the initial total? Justify your prior intuition.
2. b) Analytically verify your estimate: calculate the exact number of PINs in the system with repetition ($VR_{6,4}$) and in the system without repetition ($V_{6,4}$). Determine the percentage that PINs without repetition represent with respect to the total with repetition and evaluate whether your initial estimate was correct.

To improve the security of smart lockers at a school, the system requires a $4$-digit access code. However, to avoid obvious combinations, the system imposes the following conditions: the first digit must be odd, the last digit must be even (considering $0$ as even), and no digit may be repeated in the code. How many different codes can be generated following these rules?

In a regional E-sports competition (competitive video games), $12$ teams are participating. The organisers must award a gold, silver and bronze medal to the top three teams.

1. a) In how many different ways can the competition podium be formed?
2. b) If we know that the local team "Los Gamers" always finishes in the top three, in how many ways can the podium be completed now?

A streetwear brand wants to launch a limited edition of hoodies. They have $3$ identical red hoodies, $2$ identical blue hoodies and $4$ identical black hoodies. For a photoshoot, they must place all these hoodies in a row on a counter. In how many different ways can the hoodies be arranged based solely on their colour?

A group of $15$ students in $2º$ Baccalaureate is organising the graduation party. They must choose a committee formed of $4$ people to be in charge of the music and food.

1. a) How many different committees can be formed?
2. b) If there are $8$ girls and $7$ boys in the group, how many of those committees would be formed of $2$ girls and $2$ boys?

Seven friends go out for dinner at a pizzeria and sit around a circular table.

1. a) In how many different ways can they sit around the table?
2. b) Two of the friends, Ana and Javi, have had an argument and do not want to sit together. In how many ways can the seven friends sit now, satisfying this condition?

A café offers a "Student Deal" that consists of choosing one type of sandwich, one side dish and one drink. They have $5$ types of sandwiches, $3$ types of sides and $4$ types of drinks.

1. a) How many different meals can be created?
2. b) If a student decides that they do not want a vegetable sandwich (there is 1 type) and do not want to drink water (there is 1 type), how many options do they have now for their meal?

A courier needs to deliver a package in a city whose street layout is a perfect grid. They must travel from point $A$ (bottom left corner) to point $B$ (top right corner), moving only right or up. If the distance is $6$ blocks to the right and $4$ blocks up:

1. a) How many different routes can they take?
2. b) How many of those routes pass through a checkpoint $C$ located $3$ blocks to the right and $2$ up from point $A$?

Find the value of $n$ (an integer with $n \ge 3$) in the following equation with binomial coefficients, explaining the steps you take:

In a secondary school, a student debate committee is being organized, and three people must be chosen from the students of a 2nd year of Bachillerato group to fill three distinct roles: spokesperson, secretary, and moderator. The same person cannot hold more than one role simultaneously. After carrying out the process, the tutor confirms that there are exactly $990$ different ways to form said committee.

1. a) Model the situation by setting up a combinatorial equation in terms of the total number of students $n$ in the class, justifying the choice of the counting method used.
2. b) Solve the equation in a reasoned manner to determine how many students make up the class. To do so, factorize the number $990$ and identify the appropriate product of consecutive integers, justifying the result without resorting to blind trial and error.

The vehicle registration system in Spain assigns each vehicle a license plate with a $4$-digit block followed by a $3$-letter block. The numbers range from $0000$ to $9999$ (that is, $10$ possible digits per position, allowing repetitions). For the letter block, $20$ consonants of the Spanish alphabet are used, having excluded the five vowels (to avoid vulgar words) and the letters Ñ and Q (to avoid optical confusion with N, O, and 0). The letters can also be repeated.

1. a) Model and calculate the total number of distinct license plates that this system can issue using the multiplication principle and variations with repetition.
2. b) If in Spain an average of approximately $1,1$ million passenger cars and light vehicles are registered each year (according to DGT records), model how many full years this system will be able to operate without being exhausted or modifying its format.
3. c) Suppose that the DGT decided to add two exclusion conditions: not allowing license plates that have all $3$ letters identical (such as BBB, CCC, etc.) nor license plates whose $4$ numbers are equal ($0000, 1111, \dots, 9999$). Calculate in a reasoned manner how many valid license plates would remain under these restrictions.

In the binomial expansion of the expression $(\frac{x^2}{2} - \frac{k}{x^3})^{15}$, it is known that the coefficient of the term containing $x^{10}$ is equal to $3003$. Determine, giving reasons, the value of the positive real number $k$ and, once found, calculate the value of the independent term of this expansion, if it exists.

Determine all values of $n \in \mathbb{N}$ that satisfy the following algebraic equality based on combination numbers:

Justify the process by expanding the factorials and solving the resulting equation.

A cybersecurity company wants to generate unique access codes for its employees. Each code must be 9 characters long, distributed as follows:

• 3 initial uppercase letters (chosen from the 26 letters of the alphabet, excluding Ñ).

• 4 central numerical digits (0-9).

• 2 final special symbols (chosen from a set of 5 symbols: @, #, %, &, *).

Calculate how many different codes can be generated if the following restrictions are imposed:

1. Uppercase letters cannot be repeated.

2. Numerical digits must all be distinct and arranged in strictly increasing order.

3. Special symbols may be repeated.

An architecture firm is designing a housing estate with 7 identical plots in a row. They must plant trees of three different species: oak, pine and holm oak. For aesthetic and biological reasons, the following requirements must be met:

• Only one tree is planted in each plot.

• There must be at least one tree of each species.

• No two pine trees can be in adjacent plots.

Calculate the total number of different ways in which the trees can be distributed among the 7 plots.

A programmer is developing an algorithm to find the shortest route in a sensor network represented by a grid of $6 \times 5$ nodes. The algorithm must calculate how many paths of minimum length exist from node $A(0,0)$ to node $B(6,5)$, taking into account that due to a fault, the central node $C(3,3)$ is inaccessible and cannot be part of any route.

A debate committee of 6 students is to be formed for a Model United Nations (MUN). The available candidates are 8 students from 2nd year Baccalaureate and 6 students from 1st year Baccalaureate.

Calculate how many ways the committee can be chosen if these two conditions must be met simultaneously:

1. The committee must contain more 2nd year than 1st year Baccalaureate students.

2. There must be at least one student from each year in the selected group.

Six friends (3 boys and 3 girls) sit down to dinner at a circular table to celebrate the end of their exams. Determine the number of possible seating arrangements if:

1. There are no gender restrictions.

2. The boys and girls must sit alternately (no boy next to another boy and no girl next to another girl).

3. A specific pair of friends, Elena and Marcos, insist on sitting together.

(Note: Consider that in circular permutations, only the relative position of one person to another matters).

A streaming platform uses an algorithm to recommend playlists. If they have 10 Rock songs, 8 Pop songs, and 6 Indie songs, how many 5-song playlists can be created such that each playlist contains at least one song from each of the three musical genres? Note that the order of the songs in the playlist does matter.

A cybersecurity department generates $5$-character passwords formed exclusively with the digits from the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$. To ensure that they do not follow uniform dispersion patterns, the protocol requires that the password contain at least one repeated digit (that is, not all its digits are distinct).

To calculate how many valid passwords exist, two team members propose different strategies:

• Sandra proposes the "Direct method": classify the passwords according to the exact type of repetition (exactly one pair of equal digits, two pairs, three of a kind, etc.) and add up the cases.

• Tomás proposes the "Complementary method": calculate the total number of possible $5$-digit passwords and subtract those in which no digit repeats.

1. a) Critically compare both methods: justify in detail why the direct method is excessively complex and prone to overlapping or omission errors, while the complementary method simplifies the reasoning.
2. b) Apply the complementary method using the relevant combinatorial formulas ($VR$ and $V$) to obtain the exact number of valid passwords.
3. c) Suppose that an extra restrictive condition is added: the password must necessarily start with an odd digit (while maintaining the requirement of containing at least one repeated digit). Model and calculate in a reasoned manner the new total number of available passwords.

To access a digital platform, a $4$-character password must be designed using only the letters from the set $\{A, B, C, D, E\}$, allowing any letter to be repeated. The essential condition is that the password must contain the letter A at least once.

Javier presents the following solution in class:

"First I choose one of the $4$ possible positions to place the mandatory letter A, which gives $4$ options. Then, in each of the remaining $3$ positions, I can place any of the $5$ letters of the set without any restriction, which gives $VR_{5,3} = 5^3 = 125$ options. By the counting principle, the total number of valid passwords is $4 \cdot 125 = 500$".

However, Javier's result is incorrect due to overcounting.

1. a) Formally prove Javier's mistake: choose a specific password containing more than one letter A (for example, AABB or AAAB) and explain in detail why Javier's procedure counts it more than once, identifying at which points in his count the duplication occurs.
2. b) Correct the calculation by solving the problem using the complementary event method and check by how much the result was overestimated.
3. c) If the system's condition changed to require that the password contains exactly two letters A (allowing the other two positions to be filled with any of the remaining letters $\{B, C, D, E\}$, repeated or not), model the situation by combining independent stages or permutations with repetition and calculate the exact number of valid passwords.