Exercises: Variations

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

We roll a traditional cubic die with $6$ faces $4$ 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 movie theater and find a row with $4$ consecutive free seats. In how many different ways can they sit to watch the movie?

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 $7$ artworks are going to be exhibited. 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 organized?

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

A reading club has $9$ members. They must elect a board of directors consisting of a president, a vice president, and a secretary. Since the roles are different and each person can only hold one, in how many ways can the board 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 marked. If a student answers all the questions, in how many different ways can the exam be filled in?

The license plate system of a country is formed by a block of $4$ digits (from $0$ to $9$) followed by a block of $2$ letters chosen from an alphabet of $20$ consonants. Repetition of characters is allowed in both blocks. How many different license plates can be issued in total with this format?

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

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 start exactly with the sequence $10$?

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

How many numbers of $4$ digits can be formed with the digits $\{1, 2, 3, 4, 5\}$ (allowing repeated digits) that are strictly greater than $3000$?

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

A bicycle security padlock 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 where the three digits must be strictly different.

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 chess tournament is played as a double round-robin (each player faces everyone else twice, alternating the color of the pieces). If it is known that a total of $90$ games have been played, set up the corresponding equation using the formula for permutations and find how many players took part in the tournament.

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

A palindromic number is one that reads the same from left to right as from right to left (for example, $35853$). How many palindromic numbers of $5$ digits can be formed using only the digits from $1$ to $9$?

To access a secure database, a password of $5$ characters is required. For security, the system requires the first two characters to be vowels (repetition is allowed) and the last three characters must be numbers from $0$ to $9$ (with no repetition). How many different passwords meet this format?

A class consists of $6$ boys and $5$ girls. A committee consisting of a president, vice president, and secretary is to be chosen. The presidency must be held by a girl, the vice presidency by 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 colors available. In how many ways can the flag be colored if the only condition is that two adjacent stripes (next to each other) cannot have the same color?

Knowing that the number of permutations with repetition of $x$ elements taken $2$ at a time exceeds by $12$ the number of permutations without repetition of the same $x$ elements taken $2$ at a time, set up the corresponding mathematical equation and find the value of $x$.

Five people board a train car in which there are $8$ free seats. Two of these people are a couple and insist on sitting in a specific pair of adjacent seats next to the window. Knowing that the couple cares about who sits by the window and who sits by the aisle, in how many different ways can the five people occupy the seats?