Exercises: Combinatorics

A restaurant offers $4$ starters, $5$ main courses and $3$ desserts on its set menu. Calculate how many different complete meals (consisting of a starter, a main course and a dessert) a customer can order.

Calculate how many different ways $6$ different textbooks can be arranged in a line on a shelf.

A robotics club has $12$ members. Determine how many different ways a committee consisting of a president, a vice president and a secretary can be chosen, given that no person can hold more than one position.

A technology teacher needs to select $4$ students from a class of $20$ to attend a science fair. Calculate how many different groups of representatives can be formed.

Calculate the number of different anagrams, whether meaningful or not, that can be formed using all the letters of the word $INGENIERIA$.

A digital bicycle lock is opened using a $4$-digit code (from $0$ to $9$). Determine how many different codes can be set if repeating digits is allowed.

$8$ points are marked on a plane, no three of which are collinear. Determine how many different straight lines can be drawn by connecting pairs of these points.

From a Spanish deck of $40$ cards, $3$ cards are drawn at random without replacement. Calculate the probability that all three cards are Kings.

In a 1st year Bachillerato class with $20$ students, a group of $3$ solidarity delegates must be chosen to organise a regional food drive, such that all three members carry out exactly the same roles within the group. Marcos states: "Since there are $20$ options for the first delegate, $19$ for the second, and $18$ for the third, the total number of possible groups is $V_{20,3} = 20 \cdot 19 \cdot 18 = 6840$". Explain in detail why Marcos's reasoning is flawed, identify which combinatorial concept he failed to take into account, and calculate the actual number of different groups that can be formed.

In a regional padel league, $8$ pairs have registered. In the first stage, the organisers establish a single round-robin tournament (each pair plays against each of the others exactly once).

1. a) Before doing any exact calculation, make an intuitive estimate: do you think the total number of required matches will be closer to $25$, to $50$, or to $100$? Write down your initial estimate and briefly justify your intuition.
2. b) Model the situation by identifying the appropriate combinatorial technique and calculate the exact number of matches that will be played.
3. c) Compare the value obtained with your initial estimate and reflect on why intuition often tends to overestimate the result in pairing count problems.

Seven friends go to the cinema and sit in a row of $7$ seats. Two of them, Ana and Bruno, must sit next to each other. Calculate how many different ways they can occupy the seats.

A biomedical research team must consist of $5$ people. The available candidates are $6$ biologists and $4$ chemists. Determine how many different teams can be formed if the team must contain exactly $3$ biologists and $2$ chemists.

A batch of $20$ microchips contains $4$ defective ones. If $3$ microchips are selected at random without replacement for inspection, calculate the probability that exactly $1$ of them is defective.

A security system requires a password consisting of $2$ distinct vowels followed by $3$ distinct digits (from $0$ to $9$). Calculate how many different passwords can be configured under these conditions.

The letters of the word $AEROPUERTO$ are shuffled at random to form a new anagram. Calculate the probability that the new word starts and ends with the letter $O$.

In a chess tournament where each participant plays exactly one match against every other player, a total of $105$ matches were played. Determine the number of players who participated in the tournament.

From a standard deck of $52$ cards, a hand of $5$ cards is dealt. Calculate the probability that the player gets exactly $3$ Aces in their initial hand.

An urn contains $5$ red balls, $4$ blue balls and $3$ green balls. If $3$ balls are drawn simultaneously at random, calculate the probability that they are all of different colours.

A regional medical emergency centre organises night shifts in which each duty shift must be covered by a pair of $2$ doctors. Currently, by combining the doctors on the available staff, exactly $66$ different on-call pairs can be formed (regardless of the order in which they are assigned within the pair).

1. a) Set up an equation using the corresponding combinatorial formula and solve for how many doctors make up the current staff.
2. b) Looking ahead to the summer season, the management estimates that they will need to have at least $105$ different on-call pairs to avoid work overload and ensure adequate rotation. How many additional doctors, at a minimum, will the regional centre need to hire?

The standard vehicle registration system in Spain (DGT) assigns each vehicle a license plate consisting of a block of $4$ numerical digits followed by a block of $3$ letters. For the numbers, digits from $0$ to $9$ are used and can be freely repeated (from 0000 to 9999). For the letters, $20$ consonants of the alphabet are used (the five vowels are excluded to avoid acronyms or inappropriate words, as well as the Ñ and Q to avoid visual confusion with N and O or 0), and repetition is also allowed.

1. a) Model the number of combinations provided by the numeric block and the alphabetic block using appropriate combinatorial tools, and deduce the total number of different license plates this system allows.
2. b) According to registration statistics from the DGT and automotive industry associations, around $1{,}2$ million vehicles requiring this type of plate are registered annually in Spain. Assuming a constant annual rate, estimate how many years it will take for the system to exhaust all its plates and assess whether the design is suitable for the Spanish vehicle fleet in the medium and long term.

A tech company is going to form a cybersecurity committee of $6$ experts chosen from $8$ software engineers and $7$ network analysts. Calculate how many different committees can be formed if there must be at least $4$ software engineers in the group.

Solve the following combinatorial equation to find the natural value of $n$:

$3 \cdot V_{n, 3} = 4 \cdot V_{n-1, 3}$

Four men and four women are sitting down to dinner around a circular table. Determine how many different ways they can be seated if people of the same gender cannot sit next to each other (that is, men and women must alternate). Consider two arrangements to be identical if one can be obtained by rotating the other.

In a quality control process, a random sample of $5$ items is taken from a production line containing a total of $50$ items, $10\%$ of which are defective. Calculate the probability of finding at most $1$ defective item in the sample taken.

An investor wishes to divide $10\,000$ euros among $4$ start-ups. If the investment in each company must be an exact multiple of $1\,000$ euros (including the possibility of investing $0$ euros in any of them), determine how many different ways the capital can be distributed.

$5$ cards are drawn at random from a standard deck of $52$ cards. Calculate the probability of obtaining a full house, that is, a hand consisting of $3$ cards of one rank and $2$ cards of another rank.

In a city whose streets form a perfect grid, calculate how many paths of minimum length exist to go from point $A(0,0)$ to point $B(6,5)$, passing through intersection $C(3,2)$.

The European Space Agency needs to assign $12$ astronauts to $3$ different space modules (Command Module, Lunar Module and Service Module). Calculate how many different ways the crew members can be distributed if each module must accommodate exactly $4$ astronauts.

A technical committee of the Department of the Environment must be made up of $5$ specialists chosen from a group of candidates consisting of $6$ biologists, $5$ chemists, and $4$ legal experts. By requirement of the call, the committee must include at least one legal expert. To calculate the total number of possible committees, three panel members propose different strategies:

• Strategy A (Direct method): Break down the count into four disjoint cases depending on whether the committee has $1$, $2$, $3$, or $4$ legal experts, calculate the possible combinations in each case, and add them together.

• Strategy B (Indirect or complementary method): Calculate the total number of $5$-person committees that can be formed from the $15$ candidates without any preconditions, and subtract those committees in which there is no legal expert.

• Strategy C (Pre-assignment): First choose $1$ of the $4$ available legal experts to satisfy the legal requirement, and then fill the remaining $4$ spots by freely choosing from the $14$ remaining candidates.

1. a) Critically analyse the three strategies: demonstrate which of them commits a serious conceptual counting error (explaining which elements are overcounted) and compare the computational efficiency between the two valid ones, justifying which is preferable.
2. b) Apply the most efficient valid strategy to calculate the exact number of valid committees that can be formed.

A bicycle courier needs to travel across the regular grid layout of the Eixample district in Barcelona. They must depart from the central warehouse and deliver an urgent parcel to a location situated exactly $6$ blocks East and $4$ blocks North. For the route to be of minimum distance, at each intersection they can only pedal East (E) or North (N). However, the City Council has closed the intersection located $3$ blocks East and $2$ blocks North of the warehouse to traffic due to roadworks.

1. a) Rigorously justify why the calculation of the total number of theoretical minimum routes (without considering the roadworks) can be modelled using both permutations with repetition and combinations, and calculate that total number.
2. b) Mathematically model the number of routes that necessarily pass through the intersection closed due to roadworks, dividing the journey into two consecutive independent stages and applying the multiplication principle.
3. c) Deduce the final number of different optimal routes available to the courier to complete the delivery while avoiding the roadworks area.

Unit challenge. When will the current number plate system run out in Spain? Your mission is to advise traffic authorities by analyzing the expiration of the current model and designing a viable alternative.

1. Data search: Consult the INE or DGT portal for the number of vehicle registrations in Spain over the past year (approximately $1,1$ million).

2. Combinatorial analysis: The current standard system consists of $4$ digits ($0$ to $9$) and $3$ letters ($20$ consonants are used, omitting vowels, Ñ, and Q). Calculate the total number of possible number plates using permutations with repetition ($VR_{n,k}$) and the multiplication principle.

3. Spreadsheet simulation: Model the evolution of the vehicle fleet in a spreadsheet with the INE data to project how many years of validity remain for the system before collapsing.

4. Technical proposal: Design a new number plate format that multiplies the current capacity by at least $4$. Calculate its combinatorial sample space and present your conclusions in a brief one-page report.

Assessment criteria:

• Identification and correct application of combinatorial tools (permutations, multiplication principle).

• Rigour in obtaining, processing, and interpreting real INE data in the spreadsheet.

• Feasibility, creativity, and mathematical foundation of the newly proposed system.