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 theatre 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: $\frac{(n+2)!}{n! \times (n+1)}$.

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.

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.

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?

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...)?

An Post delivery van must navigate through the grid layout of the city centre in Dublin. It starts from a warehouse and must reach a delivery point located 4 blocks to the east and 3 blocks to the north. The van always follows paths of minimal length, moving exclusively east ($E$) or north ($N$) in each section.

1. a) Model this situation using permutations with repetition, justifying why each possible route is equivalent to a sequence of letters of fixed length, and calculate the total number of distinct minimal paths.
2. b) If the intersection located 2 blocks east and 1 block north of the origin is completely closed due to municipal roadworks, calculate how many minimal paths remain available that avoid passing through that intersection. Justify the procedure you followed.

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.

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 $x$: $\frac{(x+1)!}{(x-1)!} - \frac{x!}{(x-2)!} = 14$.

In a scientific symposium organized at University College Dublin (UCD), 6 speakers participate: 3 physics specialists and 3 mathematics specialists. They will speak consecutively in the same room, and the organizing committee requires that no two physicists speak in succession. To determine how many valid schedules exist, two students propose different approaches:

• Aoife proposes the complementary method: "We calculate all possible arrangements of the 6 speakers and subtract those in which the physicists appear together."

• Cian proposes the gap method: "We first place the 3 mathematicians in a row, identify the available gaps between them and at the ends, and place the physicists there."

1. a) Analyze both proposals and explain precisely why Aoife's strategy is extremely complex and error-prone in this specific case (point out what overlapping or different cases she would have to distinguish when trying to subtract).
2. b) Fully develop Cian's strategy by breaking down the factors using permutations, calculate the final result, and conclude by justifying why this method is safer and more efficient.

At a diplomatic summit held in Kilkenny, a dinner is organized for 8 leaders at a circular table with 8 seats, where only the relative position of the diners with respect to each other is taken into account. Protocol imposes two simultaneous restrictions:

1. The leaders of Spain and Portugal must sit together, side by side.

2. The leaders of France and Germany must sit in diametrically opposite positions (directly facing each other across the center of the table).

Design a rigorous strategy to count the possible configurations by fixing one diner as a reference to break the circular symmetry, analyze in detail how many compatible positions remain for the Spanish-Portuguese block and for the rest of the delegations, and deduce the total number of valid ways to seat the 8 leaders.