Exercises: Combinatorics

A 6th Year Leaving Certificate student wants to set up a $4$-digit access code with different digits for their gym locker. They decide to use only the odd digits $\{1, 3, 5, 7, 9\}$ and cannot repeat any digit.

1. a) How many different codes can they create?

1. b) If the student forgets the code but remembers that the first digit was a $7$, how many different codes would they have to try at most?

In a robotics championship, a team must select $3$ representatives from a group of $8$ students to present the project to the panel of judges.

1. a) In how many different ways can the group of representatives be chosen if the selection order does not matter?

1. b) If among the $8$ students there are $2$ who specialise in programming and it is decided that at least one of them must be in the group, how many different selections are possible?

Let $A$ and $B$ be two events associated with a random experiment such that $P(A) = 0.6$, $P(B) = 0.4$ and $P(A \cap B) = 0.2$. Calculate the following probabilities:

1. a) $P(A \cup B)$

1. b) $P(\bar{A} \cap \bar{B})$ (probability that neither $A$ nor $B$ occurs).

1. c) $P(A - B)$ (probability that $A$ occurs but $B$ does not occur).

The Venn diagram shows the events $A$ and $B$ within the sample space $E$, with the probability of the intersection.

In a study on the use of mobile devices in a class of $30$ students, the following contingency table was obtained, classifying students by their preferred operating system and whether they regularly shop online:

If a student is chosen at random, calculate:

1. a) The probability that they use iOS.

1. b) The probability that they make online purchases given that they use Android.

1. c) Are the events "Using Android" and "Making online purchases" independent? Justify your answer.

A video game server has two connection nodes: Node 1, which serves $70\%$ of the users, and Node 2, which serves the rest. It is known that $5\%$ of the connections on Node 1 suffer lag, while on Node 2 this problem occurs in $10\%$ of the connections.

1. a) Represent the situation using a probability tree diagram.

1. b) If a player joins the server at random, what is the probability that their connection suffers lag?

In an electronics factory, the probability that a microchip is defective is $0.02$. If the microchips are packed in boxes of $10$ units and the presence of defects in one is independent of the others:

1. a) What is the probability that there are no defective microchips in a box?

1. b) What is the probability that there is exactly one defective microchip in the box? (Express the result using binomial coefficients).

Three phone operators, $M_1$, $M_2$ and $M_3$, handle $40\%$, $35\%$ and $25\%$ of customer service call centre calls, respectively. The percentage of calls that drop unexpectedly for each operator is $2\%$, $3\%$ and $4\%$, respectively.

A call is chosen at random and turns out to have dropped. Calculate the probability that it was handled by operator $M_1$.

Use the properties of binomial coefficients and the Binomial Theorem to solve:

1. a) Fully expand the expression $(x + 2)^4$.

1. b) Without expanding the whole binomial, calculate the value of the coefficient of the term containing $x^3$ in the expansion of $(x + 3)^5$.

The figure shows the rows of Pascal's triangle up to $n = 5$. Its numbers are the binomial coefficients $\binom{n}{k}$: remember that the coefficient of a term of the binomial, besides that number, also includes the powers of the second term.

To elect the student delegation that will represent a high school on the local Student Council, $3$ students must be selected from a group of $24$ volunteers from 6th Year. Cian reasons as follows on his exam: «For the first spot I have 24 options, for the second 23, and for the third 22, so the total number of possible delegations is $V_{24,3} = 24 \times 23 \times 22 = 12\,144$».\n\na) Identify the conceptual error made by Cian based on the two key questions of combinatorics: does order matter in this case? Can elements be repeated?\nb) Correct the calculation and find the real number of possible delegations, justifying which operation Cian's formula must be replaced with.\nc) If the three spots had differentiated roles (a spokesperson, a secretary, and a committee member), would Cian's original calculation be correct? Justify your answer.

A municipal electric bicycle rental service in an Irish city is renewing its smart locks. Each user must set a $4$-digit PIN code with two conditions: the first digit cannot be $0$, and no two consecutive digits can be the same (for example, the code $4727$ is valid, but $4772$ is not).\n\na) Before performing any mathematical calculation, intuitively estimate whether the number of valid codes will be closer to $5\,000$, to $6\,500$, or to $9\,000$. Briefly justify the basis of your estimate.\nb) Apply the fundamental counting principle to calculate exactly how many different codes satisfy both conditions.\nc) Check whether the exact result obtained supports your initial estimate, and calculate what percentage it represents relative to the total number of four-digit combinations that do not start with zero (from $1\,000$ to $9\,999$).

A streaming platform is designing a new custom interface. For the user profile, an avatar can be chosen by combining the following elements: 4 different face shapes, 5 types of hairstyle and 3 styles of glasses. In addition, the username must be a 5-character string consisting of 3 lower-case letters (from a 26-letter alphabet) followed by 2 digits.

1. a) How many different avatars can be created?
2. b) How many distinct usernames can be generated if neither letters nor digits can be repeated?
3. c) If a user decides to choose their avatar and username at random (under the conditions of part b), what is the probability that their name starts with the letter "z" and ends with the digit "9"?

In a robotics competition, teams must pass two independent tests. It is known that the probability of a robot passing the speed test is $P(V) = 0.75$ and the probability of passing the accuracy test is $P(P) = 0.60$. Calculate the probability that a randomly chosen team:

1. a) Passes at least one of the two tests.
2. b) Passes the speed test but not the accuracy test.
3. c) Given that it has passed at least one test, what is the probability that it has passed both?

A cybersecurity system uses two filters to detect phishing emails. Filter A detects $92\%$ of these emails, whilst Filter B detects $88\%$. Due to the structure of the system, an email is analysed by Filter B only if it has not been detected by Filter A. If an email is not detected by either filter, it reaches the user's inbox. It is estimated that $3\%$ of the emails received in the company are phishing attempts.

1. a) What is the probability that a phishing email reaches the inbox?
2. b) If an email has reached the inbox, what is the probability that it is actually a phishing email?

A 3D printing electronics manufacturing company has observed that the probability of a part having a finishing defect is $5\%$. If a batch of 12 parts is printed:

1. a) What is the probability that exactly 2 parts are defective?
2. b) What is the probability that there is more than one defective part?
3. c) Calculate the mean and standard deviation of the number of defective parts in the batch.

The range of a new fleet of electric scooters for urban use follows a normal distribution with mean $35$ km and standard deviation $4$ km.

1. a) Calculate the probability that an e-scooter chosen at random has a range greater than $40$ km.
2. b) What percentage of e-scooters will have a range between $30$ and $38$ km?
3. c) If the company wants to ensure that $90\%$ of its e-scooters exceed a certain minimum range $d$, what must be the value of $d$?

A survey of 250 students at a technology centre was carried out regarding their use of operating systems. The results are shown in the following contingency table:

If a student is chosen at random, calculate:

1. a) The probability that they are a Higher Vocational student and mainly use Linux.
2. b) Given that the student mainly uses Windows, what is the probability that they are an Intermediate Vocational student?
3. c) Are the events "being a Higher Vocational student" and "mainly using Linux" independent? Give reasons for your answer.

It is known that $15\%$ of users visiting a computer components retail website end up making a purchase. If on a given day the website receives 400 independent visits:

1. a) State which probability distribution the variable "number of purchases made" follows and justify whether it can be approximated by a normal distribution.
2. b) Using the appropriate approximation, calculate the probability that between 50 and 70 purchases are made (inclusive).

A laboratory has 8 precision temperature sensors and 6 humidity sensors. For an environmental control experiment, 4 sensors must be selected at random.

1. a) In how many different ways can the selection of 4 sensors be made if order does not matter?
2. b) What is the probability that exactly 2 temperature sensors and 2 humidity sensors are selected?
3. c) What is the probability that all selected sensors are of the same type?

The chess federation of a county is organising a preliminary knockout stage in which all registered participants must play exactly one match against each of the other opponents ("round-robin"). The booked venue and available schedule only allow for a maximum of $100$ matches in total.\n\na) Set up an inequality in terms of the number of players, $n$, that models the total number of matches in the tournament. Justify which combinatorial structure represents this number of matches.\nb) Solve the inequality to determine the maximum number of participants that can register without exceeding the match limit, and calculate how many matches would be played exactly with that maximum number.\nc) If $8$ participants ultimately register and the organisation decides to award a gold, a silver, and a bronze trophy to the top three finishers (no ties), how many possible podium allocations are there?

To grade the Leaving Certificate Higher Level Mathematics exams at a State Examinations Commission centre, an examining board of $6$ examiners must be formed. The available faculty consists of $10$ teachers: $6$ come from non-fee-paying secondary schools and $4$ from fee-paying private secondary schools. The organising committee's regulations require that the board have at least $2$ teachers from non-fee-paying schools and at least $2$ from fee-paying private schools.\n\na) Model the situation by identifying all valid cases for the composition of the board according to the number of teachers from each type of school.\nb) Calculate how many different boards can be formed in compliance with the established regulations.\nc) If the $6$-person board were chosen completely at random from the $10$ teachers (without requiring any representation quota), what would be the probability that the board complies with the legal regulations? Interpret whether the restriction excludes many or few of the possible combinations.

A cybersecurity team is analysing the vulnerability of an access system that uses passwords of $8$ characters. The allowed characters are the $26$ letters of the alphabet (case-insensitive) and the digits from $0$ to $9$. A password is considered "strong" if it meets the following conditions:

1. It contains exactly $2$ distinct numbers and $6$ letters.

2. The letters cannot be repeated.

3. The two numbers must be in consecutive positions.

Calculate:

1. a) The total number of strong passwords that can be generated.
2. b) If brute-force software generates a random password following only the structure of $8$ characters (letters or numbers), what is the probability that it guesses a specific strong password on the first attempt?

In the expansion of a new binomial, an engineering student needs to determine certain coefficients. Consider the binomial expansion of $Q(x) = \left( \dfrac{x^2}{2} - \dfrac{k}{x} \right)^{12}$, where $k$ is a positive real number.

1. a) Determine the value of $k$ knowing that the term independent of $x$ in the expansion of $Q(x)$ is equal to $7920$.
2. b) Having found $k$, calculate the coefficient of the term containing $x^9$.

A streaming platform uses a recommendation algorithm to predict whether a user will like a new series. The algorithm sometimes fails: $10\%$ of the series that a user would love are not recommended ($FN$: false negative), and $20\%$ of the series that a user would dislike are mistakenly recommended ($FP$: false positive).

If $60\%$ of all series in the catalog are potentially to the taste of a specific user, calculate:

1. a) The probability that the algorithm recommends a randomly chosen series.
2. b) If the algorithm has just recommended a series to this user, what is the probability that it is indeed a series they like?

A courier company uses drones to deliver packages in a city. The probability of a drone suffering a technical breakdown during a delivery flight is $0.02$. The company makes $250$ daily deliveries independently.

1. a) Justify whether the variable "number of daily breakdowns" can be approximated by a normal distribution. If so, state its parameters.
2. b) Using the appropriate approximation, calculate the probability that on a given day exactly $5$ breakdowns occur.
3. c) Calculate the probability that between $3$ and $8$ breakdowns occur (inclusive).

In a genetics laboratory, the resistance of a plant to a pest is studied. It is known that the probability of a plant being resistant is $p$. $8$ plants are chosen at random and it is observed that the probability that exactly $2$ are resistant is equal to the probability that exactly $3$ are resistant.

1. a) Calculate the value of the probability $p$.
2. b) With the value of $p$ found, calculate the probability that at least one plant is resistant.
3. c) Determine the expectation and standard deviation of the number of resistant plants in a sample of $100$ plants.

The weight of coffee packets produced by a factory follows a normal distribution $N(\mu, \sigma)$. It has been observed that $15.87\%$ of the packets weigh more than $510$ grams, and that $2.28\%$ of the packets weigh less than $480$ grams.

1. a) Calculate the mean $\mu$ and standard deviation $\sigma$ of the distribution. (Recall that for a $Z \sim N(0,1)$, $P(Z < 1) \approx 0.8413$ and $P(Z < 2) \approx 0.9772$).
2. b) If the net weight indicated on the label is $500$ grams, what percentage of the packets is below the legal weight?

A competitive examination consists of $80$ topics. The candidate studies $55$ topics chosen at random. The examination board draws $3$ topics by lottery and the candidate must choose one to write on.

1. a) What is the probability that the candidate does not know any of the three drawn topics?
2. b) What is the probability that they know exactly two of the topics?
3. c) If the candidate passed the examination (which implies they knew at least one of the topics), what is the probability that they knew all three drawn topics?

An electronic components factory has three production lines: $L_1$, $L_2$ and $L_3$. Line $L_1$ produces twice as much as $L_2$, and $L_3$ produces the same amount as $L_1$ and $L_2$ combined. The percentage of defective components is $1\%$ in $L_1$, $3\%$ in $L_2$ and $2\%$ in $L_3$.

1. a) Calculate the probability that a component chosen at random from the total production is defective.
2. b) If we select a component and it turns out to be defective, what is the probability that it comes from line $L_3$?
3. c) If $10$ components from line $L_1$ are analysed independently, what is the probability that at least one of them is defective?

At the 6th Year graduation ceremony, $8$ honours awards are presented. On stage, $8$ seats have been arranged in a line and numbered from $1$ to $8$. Among the $8$ award-winning students are Aoife and Seán, who have had a recent disagreement, and the organisation wishes to seat them so that they are not in adjacent seats. Two members of the organising committee debate how to calculate the number of valid seating arrangements:\n- Method A (gap method or direct method): The other $6$ students are seated first in order, and then two of the $7$ possible gaps (ends or between peers) are chosen to seat Aoife and Seán.\n- Method B (complementary method): The total number of ways to seat the $8$ students without any restrictions is calculated, and the unfavourable cases, where Aoife and Seán sit together forming a single block, are subtracted.\n\na) Detail the combinatorial approach and the numerical result applying Method A.\nb) Detail the combinatorial approach and the numerical result applying Method B.\nc) Verify that both approaches lead to the same solution, and justify in a reasoned way which of the two methods would be clearly preferable and less prone to errors if the restriction required that three specific students (Aoife, Seán, and Cian) cannot sit in adjacent positions, none next to each other.

A vehicle registration system in Ireland assigns each vehicle a license plate consisting of a block of $4$ digits (from $0000$ to $9999$) followed by a block of $3$ consonant letters chosen from $20$ possible letters (the vowels A, E, I, O, U, and the letters Ñ and Q are excluded to avoid confusion and inappropriate combinations).\n\na) Calculate the total number of theoretical license plates that this system can issue and express the result using powers of base ten.\nb) A combinatorics enthusiast looks for "symmetric license plates": those whose $4$-digit numerical block is a palindrome (for example, $4884$ or $0330$) and whose $3$-letter block is a palindrome (the first and third letters are the same, as in $TRT$). Determine how many symmetric license plates exist and find the probability that a randomly registered vehicle has this property.\nc) The Road Safety Authority (RSA) is investigating a license plate reading algorithm and detects errors in plates that it calls "heterogeneous mixed": those that have their $4$ digits strictly distinct from each other and, simultaneously, contain exactly two identical letters and one different letter in the letter block (in any order, as in $LBL$ or $DDH$). Model and calculate how many "heterogeneous mixed" license plates exist in the potential Irish vehicle fleet.

Unit Challenge. Youth labor market audit with CSO data.

Your mission is to advise an employment committee by analyzing the relationship between educational attainment and youth unemployment using real data.

1. Data collection: Access the CSO portal (most recent Labour Force Survey, LFS) and extract the frequencies of young people (aged 16-29) in your region according to educational level (basic, secondary, higher) and employment status (employed, unemployed).

2. Structuring and counting: In a spreadsheet, build the contingency table and its tree diagram with relative frequencies. Apply combinatorics to calculate in how many different ways you could choose a sample of $5$ young people where at least $3$ have higher education.

3. Bayesian modeling: Calculate the total probability of unemployment $P(D)$ and apply Bayes' theorem to find the posterior probability $P(\text{Higher} \mid D)$. Compare the result with the prior probability $P(\text{Higher})$ and determine whether both events are independent.

4. Conclusions report: Write a reasoned report (one page) with explanatory graphs, justifying, based on your calculations, whether a higher educational level is associated with a lower risk of unemployment, and explaining why this association alone does not prove causation.

Assessment criteria:

• Rigor in using probability properties, total probability, and Bayes' theorem.

• Correctness in applying counting techniques and spreadsheet structuring.

• Quality of the report and consistency in the critical interpretation of CSO data.