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 theatre and buys tickets for consecutive seats in the same row. In how many different ways can they sit to enjoy the play?

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?

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

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

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

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

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?

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...)?

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?

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

Seán wants to calculate how many different words, whether meaningful or not, can be formed by rearranging all the letters of the word "PATATA". On his exam, he writes the following:

"Since the word has 6 letters, there are $6! = 720$ arrangements. But since the letter A is repeated 3 times and the letter T is repeated 2 times, I subtract them from the total: $6! - 3! - 2! = 720 - 6 - 2 = 712$ possible words".

Identify the conceptual error Seán has made, explain in your own words why subtracting factorials makes no sense in this context of arrangements, and calculate the correct solution by applying and justifying the appropriate formula.

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?

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

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 training deck for mental calculation contains 8 cards in total. Two of them are red, and the rest are blue cards and green cards. When all the cards are arranged in a row on the table, it turns out that exactly 420 distinct arrangements can be formed taking into account only the sequence of colors.

1. a) Set up a combinatorial equation relating the number of blue cards ($a$) and the number of green cards ($v$) to the total number of possible arrangements.
2. b) Deduce how many blue cards and how many green cards make up the deck. Justify why two symmetric numerical solutions appear and explain their practical meaning.

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 $A$?

Four friends go for an afternoon snack at a downtown cafe. They hesitate between sitting at a bar with 4 stools in a row or sitting at a circular table with 4 seats where only the relative position between them matters (who sits to the left or right of whom).

1. a) Before applying any formula, make a reasoned estimate of which of the two situations will offer a greater number of distinct arrangements and venture what approximate relationship there will be between both results.
2. b) Calculate the exact number of arrangements at the bar ($P_4$) and at the round table ($PC_4$).
3. c) Check whether the calculation confirms your initial estimate and explain why fixing one person as a reference point at the table reduces the total number of arrangements by exactly that factor.

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?

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?

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?

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?