Exercises: Permutations

Calculate the numerical value of the following expressions involving factorials: $\frac{8!}{5! \times 3!}$ and $\frac{12!}{10!}$.

A group of $6$ friends goes to the theater and buys tickets for consecutive seats in the same row. In how many different ways can they sit to enjoy the play?

How many different words, whether meaningful or not, can be formed by rearranging all the letters of the word "MUNDO"?

In an art gallery, $5$ paintings by a local artist are going to be displayed in a straight line on a main wall. In how many different ways can the works be arranged?

During a school athletics competition, $7$ runners reach the finish line with no ties. How many different finish orders can there be at the end of the race?

A music festival has scheduled performances by $4$ rock bands for Saturday night. In how many ways can the order in which they take the stage to play be organized?

To configure a secure password, an application requires using exactly the characters $A, B, C, 1, 2$, without repeating any of them. How many different passwords can be created in the system?

Simplify the following algebraic expression involving factorials, expressing the result in its simplest form: $\frac{(n+2)!}{n! \times (n+1)}$.

Five friends go to the cinema and sit in adjacent seats. Two of them, Amina and Unai, want to sit together to share popcorn. In how many ways can the five friends be seated while always fulfilling this condition?

A ship communicates with other vessels using coloured flags hoisted vertically on a mast. If it has $3$ red flags, $2$ blue flags, and $1$ yellow flag, all of the same size, how many different signals can it send using all the flags at once?

A family of $6$ members sits down to dinner around a circular table in a restaurant. In how many different ways can they be arranged if only their relative positions to one another matter and not the exact chair they occupy in the room?

How many different words, whether meaningful or not, can be formed using all the letters of the word "CAMPANA"?

In the final of a swimming race at the Olympic Games, $8$ swimmers participate. If we know with certainty which swimmer won the gold medal and which swimmer finished in last place due to an injury, in how many different ways could the other participants finish in the remaining positions?

On the map of a city with streets laid out in a grid, a bicycle courier must travel from point $A$ to point $B$, moving only to the right and upwards. How many different paths of minimum length can they take to make the delivery?

A software engineer wants to form six-digit numerical codes using exactly the following digits in each code: $2, 2, 3, 3, 3, 4$. How many different codes can the system program?

A group consisting of $3$ boys and $3$ girls are going to sit in a row of $6$ seats to watch a concert. In how many ways can they do so if they must sit such that the genders strictly alternate (boy-girl-boy... or girl-boy-girl...)?

Find the value of the natural number $x$ that satisfies the following equation, justifying your answer mathematically: $P_x = 30 \times P_{x-2}$.

How many different words, meaningful or not, can be formed by rearranging all the letters of the word "ESTADISTICA" so that they must begin with the letter $E$ and end with the letter $A$?

Eight diplomats meet at an international summit and must sit around a large round table. Two of the representatives are strongly hostile to each other, and protocol requires that they do not sit next to each other. In how many different ways can they be seated at the table fulfilling this restriction?

On a school library shelf, $4$ different Mathematics books, $3$ different Physics books, and $2$ different Chemistry books are to be placed in a row. If the librarian requires that books on the same subject must always stay together as a block, in how many different ways can they be arranged on the shelf?

An autonomous robot moves through a logistics grid from node $A$ to node $B$, moving exclusively to the right and downwards. Due to floor repairs, node $C$ is blocked and the robot cannot pass through it. How many valid paths exist to complete the route?

Calculate the total sum of all $4$-digit integers with distinct digits that can be formed using exclusively the digits $1, 2, 3$ and $4$.

If all possible words, meaningful or not, are generated by permuting the letters of the word "LIBRO" and then strictly ordered alphabetically (like in a dictionary), what exact position will the word "LIBRO" occupy in that list?

Simplify the algebraic fractions and solve the following equation containing factorials, finding the value of $x$: $\frac{(x+1)!}{(x-1)!} - \frac{x!}{(x-2)!} = 14$.