Exercises: Permutations

Calculate the numerical value of the following expressions with factorials: $\frac{8!}{5! \cdot 3!}$ 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 they make sense in the dictionary or not, can be formed by arranging all the letters of the word "MUNDO"?

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

During a school athletics competition, $7$ runners reach the finish line without any draws occurring. How many different rankings can occur at the end of the race?

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

To set up 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: $\frac{(n+2)!}{n! \cdot (n+1)}$.

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

A ship communicates with other vessels using colored 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 seated if only the relative position between them matters and not the exact chair they occupy in the room?

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

In the final of an Olympic swimming race, $8$ swimmers participate. If we know for certain which swimmer has won the gold medal and which one has come in last place due to an injury, in how many different ways could the other participants have ranked in the intermediate positions?

On a city map with streets arranged in a grid, a delivery person on a bicycle must go 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 this if they must sit so that the genders are strictly alternated (boy-girl-boy... or girl-boy-girl...)?

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

How many different words, with or without meaning, can be formed by rearranging all the letters of the word "ESTADISTICA" so that they necessarily start 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 bitter enemies 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 going to be placed in a row. If the librarian requires that books on the same subject must always remain together forming 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 $B$ advancing 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 the integers of $4$ different digits that can be formed using exclusively the digits $1, 2, 3$ and $4$.

If all possible words, with or without meaning, are generated by permuting the letters of the word "LIBRO" and then strictly ordered alphabetically (as 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 $x$: $\frac{(x+1)!}{(x-1)!} - \frac{x!}{(x-2)!} = 14$.