Arrangements (permutations of r items)

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?

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?

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?

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?

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?

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 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 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?

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?

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

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

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?

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?

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 $10$?

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 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?

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?

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?

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

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 a chess tournament, a double round-robin is played (each player faces all the others twice, alternating the color 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.

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 $x$.