Exercises: Permutations

Simplify the following expression with factorials without using a calculator: $\frac{12! \cdot 5!}{10! \cdot 7!}$.

In an athletics competition, $8$ runners participate in the final of the $100$ metres sprint. Assuming there are no ties, in how many different ways can they cross the finish line and be classified?

In a contemporary art gallery, $6$ paintings by the same artist are going to be displayed in a row along a wall. In how many different ways can the works be visually arranged?

Calculate how many different words, whether or not they make sense in Spanish, can be formed by arranging all the letters of the word $BIOLOGIA$.

A group of $5$ friends goes to a pizzeria for dinner and decides to sit around a circular table. In how many different ways can they be seated?

A security password is formed by the digits $1, 2, 3, 4, 5, 6$ and $7$, without repeating any of them. If the password must begin with the digit $7$, how many different passwords can be set up in the system?

On the mast of a sailing boat, $7$ flags are to be hoisted to transmit a signal to the shore. There are $3$ red flags, $2$ blue flags and $2$ yellow flags available, all of the same size. How many different signals can be sent using all the flags?

A local radio presenter must schedule exactly $5$ different songs during the next hour of broadcasting. In how many ways can they set the order in which the songs will be played?

At the 4th year of ESO graduation dinner, five friends are going to sit down to have dinner around a five-seat round table. They want to know how many different ways there are to arrange themselves, taking into account only who sits to the left and to the right of each person (that is, the relative order). Marcos calculates the possibilities like this: «Since there are $5$ people for $5$ seats, these are ordinary permutations: $P_5 = 5! = 120$ different ways». Explain why Marcos's reasoning is not correct for a round table, what geometric phenomenon he has not taken into account, and calculate the correct answer.

At the Sierra Nevada ski resort (Granada), the slope operators post the snow report every morning for a sector marked with $6$ consecutive markers along the run. On a specific day in February, they place exactly: $3$ markers indicating "powder snow", $2$ markers for "hard pack snow", and $1$ marker for "wet snow".

1. a) Model the situation mathematically and calculate how many different visual sequences of markers can be formed along that descent.
2. b) Explain why in the formula it is necessary to divide by $3! \cdot 2! \cdot 1!$ instead of simply calculating $6!$.

Four male friends and three female friends go to the cinema and buy tickets for seven consecutive seats in the same row. In how many ways can they sit if the four boys want to sit together and the three girls also want to sit together?

A robotic ant moves across a grid from node $A$ to node $B$. It can only move to the right or upwards following the grid lines. If to get from $A$ to $B$ it must take exactly $5$ steps to the right and $4$ steps upwards, how many different paths can it take in total?

At an end-of-year party, $6$ friends sit around a round table. Two of them, Adrián and Bruno, have fallen out and have specifically asked not to sit together. In how many different ways can the $6$ guests be arranged satisfying this condition?

Determine the numerical value of the unknown $x$ in the following algebraic equation with factorials: $\frac{(x+1)!}{(x-1)!} = 72$.

A hospital must organise the duty shifts of a doctor for a full week ($7$ days). She must be assigned exactly $3$ morning shifts, $2$ afternoon shifts and $2$ days off. In how many different ways can her weekly schedule be planned?

Numbers with $6$ digits are to be formed using exclusively the digits $3, 5$ and $8$. If the digit $3$ must appear exactly three times, the digit $5$ twice and the digit $8$ once, how many of the numbers formed are even?

On a library shelf, $4$ Mathematics books, $3$ Physics books and $2$ Biology books are to be placed. In how many ways can they be arranged if the books for each subject must be grouped together?

How many anagrams (whether or not they make sense) can be formed with all the letters of the word $PARALELOGRAMO$ in which the three letters $A$ appear consecutively forming a single block?

In the final stage of La Vuelta Ciclista a España, a decisive individual time trial takes place where the top $8$ riders in the general classification start. The organization must decide in which order these $8$ cyclists will hit the road.

1. a) Before doing any mathematical calculation, make an intuitive estimate: do you think the total number of possible starting orders will be less than $5\,000$, around $5\,000$ to $20\,000$, or will it far exceed $20\,000$ options? Write down and briefly justify your prior intuition.
2. b) Model the situation using permutations to calculate the exact value of the possible starting sequences and check whether your initial estimate was correct.
3. c) In light of the numerical result obtained, reflect: why does everyday intuition often underestimate the value of factorials so much as the number of elements increases?

In a graphic design studio in Valencia, they are creating a linear logo made up of a row of $6$ colour stripes identical in size: some are Mediterranean blue and the rest are Valencian orange. The creative team finds that, by varying the order of the stripes, exactly $15$ visually different logos can be designed.

1. a) Set up a combinatorial equality or equation relating the total number of stripes ($6$) and the factorials of the number of blue ($a$) and orange ($n$) stripes to the final result of $15$ combinations.
2. b) Deduce how many stripes of each colour make up the design (find all possible pairs of values for $a$ and $n$, keeping in mind that $a + n = 6$) and justify the result by expanding the corresponding factorials.

An automated delivery drone must travel from warehouse $A$ to customer $B$ in a city with grid-like streets, moving only North (upwards) and East (to the right). In the centre there is a rectangular park that the drone is forbidden to cross or fly over; nor may it travel along any street segment on the park's boundary (touching one of its corners is allowed). Calculate the total number of valid paths from $A$ to $B$ avoiding the park.

A necklace is to be made by stringing together $8$ spherical beads of different gemstones. Since the necklace can be fastened, rotated freely around the neck, and even turned inside out (flipped over), how many different necklace designs can be created?

An urn contains $4$ red balls, $3$ green balls and $2$ blue balls. All the balls are drawn one by one and lined up on a table. In how many different ways can they be lined up if the two blue balls cannot be in adjacent positions?

Find the value of $n$ by solving the following algebraic equation involving ordinary permutations: $P_{n} - 30 \cdot P_{n-2} = 0$.

A teacher sets her students the following probability and combinatorics challenge: "I have $5$ letters addressed to $5$ different students and $5$ envelopes with their names printed on them. If I place the letters into the envelopes completely at random, one in each envelope, in how many possible ways can it happen that exactly $4$ students receive the wrong letter and only $1$ receives the correct one?".

How many eight-digit numbers can be formed using the digits $1, 1, 2, 2, 3, 3, 4, 4$ such that all odd digits appear in the number before any even digit?

Using the letters of the word $MURCIELAGO$, how many different $10$-letter words can be formed such that the vowels always maintain their original alphabetical order (that is, $A$ before $E$, $E$ before $I$, etc.), even if they do not necessarily appear in consecutive positions?

A chess tournament is played over a maximum of $10$ games. The first player to reach $6$ wins wins the tournament (no draws are possible). If player $X$ wins the tournament on exactly the tenth game, in how many different ways could the sequence of wins and losses have developed across the $10$ games?

On a television set, a debate show is going to be recorded with $6$ guests seated in the same row of $6$ chairs: $3$ journalists and $3$ scientists. The show's production sets an essential condition: the $3$ journalists cannot sit together consecutively (all three in a single continuous block). To calculate how many valid arrangements exist, two members of the team propose different strategies:

• Method 1 (direct): Count separately the configurations in which the three journalists are completely separated from each other and those in which only two of them are together.

• Method 2 (complementary counting): First calculate all possible arrangements of the $6$ guests without any restriction and subtract those in which the $3$ journalists sit together forming a single block.

Compare both methods, justifying which of the two is more advantageous and reliable to apply in this case. Next, calculate the final result step by step using the method you selected as preferable, and justify why when counting the block of journalists you must multiply by $3!$.

At Atocha station (Madrid), a Renfe Cercanías train consists of $8$ carriages coupled in a line. In a special sustainable transport campaign, Renfe has $4$ standard passenger carriages, $2$ carriages accessible for persons with reduced mobility (PRM), and $2$ carriages with bicycle racks. Carriages of the same type are indistinguishable from one another.

1. a) Model the situation and calculate how many different compositions can be formed by joining the $8$ carriages in a row.
2. b) Due to operational reasons on the platforms, Renfe technicians decide that the two carriages with bicycle racks must necessarily be placed one in the first position (head of the train) and the other in the last position (tail of the train). Calculate how many different train formations can be assembled with this restriction.
3. c) Suppose now that the regulations change and require that the two carriages with bicycle racks must always be coupled together, one immediately behind the other, anywhere in the formation. Reason, before calculating, whether there will be more or fewer possibilities than in section b), and then determine the exact number of valid formations under this new condition.