Exercises: Arrangements

In a $100\text{ m}$ flat race, $8$ athletes participate. In how many different ways can the gold, silver, and bronze medals be awarded, knowing that there can be no ties?

We roll a traditional $6$-sided cubic die $4$ consecutive times and record the result of each roll in order. How many different outcomes can we obtain in total?

Five friends go to the cinema and find a row with $4$ consecutive empty seats. In how many different ways can they sit to watch the film?

We toss a coin $6$ consecutive times and record the exact sequence of heads and tails obtained. How many different sequences can be formed?

In an art gallery, $2$ paintings selected from a private collection of $7$ works are to be displayed. One will be placed on the left wall of the room and the other on the right wall. In how many different ways can this exhibition be arranged?

We want to paint $3$ postboxes lined up in a street. We have $4$ different colours available (red, green, blue, and yellow). If we can repeat colours on different postboxes, in how many different ways can we paint them?

A book club has $9$ members. They need to choose a committee consisting of a chairperson, a vice-chairperson, and a secretary. As the positions are distinct and each person can only hold one, in how many ways can the committee be formed?

A multiple-choice exam consists of $5$ questions. Each question has $3$ answer options (A, B, and C), of which only one can be selected. If a student answers all the questions, in how many different ways can the exam be completed?

To access her high school gym locker, Lucía must set a secret code of $3$ distinct digits chosen exclusively from prime numbers less than $10$, that is, from the set $\{2, 3, 5, 7\}$. Her classmate Marcos states: "Since the order of the digits matters and we have $4$ elements to form groups of $3$, the total number of possible codes is $VR_{4,3} = 4^3 = 64$."

1. a) Find the conceptual error made by Marcos and clearly explain which condition from the statement he ignored.
2. b) Correct his approach by deducing the appropriate formula and calculate the actual number of distinct codes that Lucía can program.

In a 4th-year ESO class with $25$ students, the class council for the school year is going to be elected, made up of three distinct positions: delegate, subdelegate, and secretary. The same student cannot hold more than one position.

1. a) Without performing the detailed calculation with the formula, make a reasoned estimate of whether the total number of possible councils will be closer to $1\,000$, $15\,000$, or $100\,000$. Briefly justify the rounding or mental approximation you used.
2. b) Model the situation by calculating the exact value using the appropriate combinatorics formula. Check whether the result confirms your estimate and explain why the order of selection is decisive in this assignment.

A country's number plate system consists of a block of $4$ digits (from $0$ to $9$) followed by a block of $2$ letters chosen from an alphabet of $20$ consonants. In both blocks, repetition of characters is permitted. How many different number plates can be issued in total with this format?

Using the digits $\{1, 2, 3, 4, 5, 6, 7\}$, how many even $3$-digit numbers can be formed if repeating digits in the same number is not allowed?

In computing, a byte is a sequence of $8$ bits (where each bit can be a $0$ or a $1$). How many different bytes exist that begin with the exact sequence $10$?

A company has $8$ employees and needs to assign $5$ different tasks to $5$ distinct employees (one task per person). If it is known that the first task must be carried out by the most senior employee, in how many ways can the tasks be distributed?

How many $4$-digit numbers can be formed using the digits $\{1, 2, 3, 4, 5\}$ (repetition allowed) that are strictly greater than $3000$?

In a basketball league, $14$ teams participate. Each team must play against all the others twice during the year (one home match and one away match). How many matches will be played in total throughout the entire season?

A bicycle security lock has $3$ independent wheels, each numbered from $0$ to $9$. Calculate the difference between the total number of possible combinations (allowing repetition) and the number of combinations in which all three digits must be strictly distinct.

In Morse code, letters are represented by sequences of dots and dashes. How many different signals can be formed using a maximum of $3$ symbols per signal?

The State Meteorological Agency (AEMET) uses identification codes to catalog its automatic weather stations in Spain. Each code consists of an initial letter identifying the river basin district ($8$ possible letters are used: B, C, D, E, M, N, S, or V) followed by a block of $4$ numerical digits.

1. a) If the $4$ digits are chosen from $0$ to $9$ and can be repeated without restriction, how many different station codes can be generated for the same river basin district? And across all $8$ river basins combined?
2. b) For a special high-mountain network, it is decided that the $4$ digits must all be distinct and none can be zero. Justify whether this new numerical block is modeled using permutations (variations) without repetition or with repetition, and calculate how many different codes remain available per district under this criterion.

The IT department of a school is creating temporary access identifiers for students. The codes are generated by arranging $k$ distinct digits chosen from the set of odd numbers $\{1, 3, 5, 7, 9\}$, without repeating any digit within the same code.

1. a) If the server has registered exactly $60$ total possible identifiers, set up an equation using the definition of permutations (variations) without repetition where the unknown is the length $k$ of the code, and find its value with reasoned steps.
2. b) If the security rules changed and repeating digits were now permitted while maintaining the same length $k$ found in the previous section, how many total codes could be created? Compare both results by calculating how many additional identifiers are gained, and explain why repetition increases the options.

In a chess tournament, a double round-robin is played (each player faces all the others twice, alternating the colour of the pieces). If a total of $90$ games were played, set up the corresponding equation using the formula for variations (arrangements) and find how many players participated in the tournament.

How many numbers with $4$ distinct digits can be formed using only the digits $\{0, 1, 2, 3, 4, 5\}$? Note that, by definition, a valid number cannot start with the digit $0$.

A palindrome number is one that reads the same forwards and backwards (for example, $35853$). How many $5$-digit palindrome numbers can be formed using only the digits from $1$ to $9$?

To access a secure database, a $5$-character password is required. For security reasons, the system requires the first two characters to be vowels (repetition is allowed) and the last three characters to be numbers from $0$ to $9$ (with none repeated). How many different passwords satisfy this format?

A class consists of $6$ boys and $5$ girls. A committee consisting of a chairperson, a vice-chairperson, and a secretary is to be chosen. The chairperson must be a girl, the vice-chairperson must be a boy, and the secretary can be any of the remaining students. In how many ways can the committee be chosen?

A flag with $4$ vertical stripes is to be designed. There are $5$ different colours available. In how many ways can the flag be coloured if the only condition is that two adjacent stripes cannot have the same colour?

Given that the number of variations with repetition of $x$ elements taken $2$ by $2$ exceeds by $12$ the number of variations without repetition of the same $x$ elements taken $2$ by $2$, set up the corresponding mathematical equation and find the value of $x$.

Five people board a train carriage in which there are $8$ empty seats. Two of these people are a couple and insist on sitting in a specific pair of adjacent seats by the window. Given that it matters to the couple who sits by the window and who sits by the aisle, in how many different ways can the five people occupy the seats?

We want to form $4$-digit numbers with distinct digits using only the digits $\{1, 2, 3, 4, 5, 6, 7\}$, with the essential condition that the resulting number must be even. Two students debate the best strategy to count them:

• Sergio's strategy (direct method): "First, we fix the last digit, which must be even ($\{2, 4, 6\}$), and then we complete the other $3$ positions by choosing and arranging digits from the remaining $6$".

• Alba's strategy (complementary method): "We calculate all possible $4$-digit numbers with distinct digits without restrictions and subtract those that end in an odd digit".

1. a) Fully work out both methods by applying the corresponding permutation (variation) formulas and verify that they yield exactly the same total number.
2. b) If we added a second simultaneous restriction: "the number must be even and also greater than $5\,000$", analyze with reasoning which of the two methods would be more convenient and describe what added difficulties the discarded method would present.

In the Debate League of an autonomous community, schools enter teams initially made up of $n$ speakers. The competition consists of two stages with different role assignments:

• In the local stage, each school must select $2$ of its $n$ speakers to perform specific and distinct roles: "opening speaker" and "closing speaker".

• In the regional stage, the organization allows the school's team to expand by adding $2$ more speakers ($n+2$ speakers in total), and $3$ distinct roles must be filled: "opening", "rebuttal", and "closing", without any student being able to take on more than one role.

The organization notes that the number of possible triads of speakers for the regional stage is exactly $12$ times the number of possible pairs for the local stage.

1. a) Model the situation by setting up an algebraic equation that relates both stages using the formula for permutations (variations) without repetition.
2. b) Solve the equation by simplifying the factorial expansions and find the possible values of $n$.
3. c) Knowing that the rules require the team to have at least $3$ speakers for the local stage to ensure substitutes ($n \ge 3$), determine with reasoning the initial number $n$ of speakers and verify that the solution satisfies the theoretical condition $k \le n$ in both stages.