Arrangements

Five friends go to the cinema and find a row with $4$ consecutive empty seats. In how many different ways can they sit to watch the film?

In a $100\text{ m}$ flat race, $8$ athletes participate. In how many different ways can the gold, silver, and bronze medals be awarded, knowing that there can be no ties?

A book club has $9$ members. They need to choose a committee consisting of a chairperson, a vice-chairperson, and a secretary. As the positions are distinct and each person can only hold one, in how many ways can the committee be formed?

We roll a traditional $6$-sided cubic die $4$ consecutive times and record the result of each roll in order. How many different outcomes can we obtain in total?

We want to paint $3$ postboxes lined up in a street. We have $4$ different colours available (red, green, blue, and yellow). If we can repeat colours on different postboxes, in how many different ways can we paint them?

A country's number plate system consists of a block of $4$ digits (from $0$ to $9$) followed by a block of $2$ letters chosen from an alphabet of $20$ consonants. In both blocks, repetition of characters is permitted. How many different number plates can be issued in total with this format?

In Morse code, letters are represented by sequences of dots and dashes. How many different signals can be formed using a maximum of $3$ symbols per signal?

To access her high school gym locker, Aoife must set a secret code of $3$ distinct digits chosen exclusively from prime numbers less than $10$, that is, from the set $\{2, 3, 5, 7\}$. Her classmate Cian states: "Since the order of the digits matters and we have $4$ elements to form groups of $3$, the total number of possible codes is $VR_{4,3} = 4^3 = 64$."

1. a) Find the conceptual error made by Cian and clearly explain which condition from the statement he ignored.
2. b) Correct his approach by deducing the appropriate formula and calculate the actual number of distinct codes that Aoife can program.

A bicycle security lock has $3$ independent wheels, each numbered from $0$ to $9$. Calculate the difference between the total number of possible combinations (allowing repetition) and the number of combinations in which all three digits must be strictly distinct.

To access a secure database, a $5$-character password is required. For security reasons, the system requires the first two characters to be vowels (repetition is allowed) and the last three characters to be numbers from $0$ to $9$ (with none repeated). How many different passwords satisfy this format?

Using the digits $\{1, 2, 3, 4, 5, 6, 7\}$, how many even $3$-digit numbers can be formed if repeating digits in the same number is not allowed?

How many $4$-digit numbers can be formed using the digits $\{1, 2, 3, 4, 5\}$ (repetition allowed) that are strictly greater than $3000$?

How many numbers with $4$ distinct digits can be formed using only the digits $\{0, 1, 2, 3, 4, 5\}$? Note that, by definition, a valid number cannot start with the digit $0$.

We want to form $4$-digit numbers with distinct digits using only the digits $\{1, 2, 3, 4, 5, 6, 7\}$, with the essential condition that the resulting number must be even. Two students debate the best strategy to count them:

• Seán's strategy (direct method): "First, we fix the last digit, which must be even ($\{2, 4, 6\}$), and then we complete the other $3$ positions by choosing and arranging digits from the remaining $6$".

• Aoife's strategy (complementary method): "We calculate all possible $4$-digit numbers with distinct digits without restrictions and subtract those that end in an odd digit".

1. a) Fully work out both methods by applying the corresponding permutation (variation) formulas and verify that they yield exactly the same total number.
2. b) If we added a second simultaneous restriction: "the number must be even and also greater than $5\,000$", analyze with reasoning which of the two methods would be more convenient and describe what added difficulties the discarded method would present.

A palindrome number is one that reads the same forwards and backwards (for example, $35853$). How many $5$-digit palindrome numbers can be formed using only the digits from $1$ to $9$?

A flag with $4$ vertical stripes is to be designed. There are $5$ different colours available. In how many ways can the flag be coloured if the only condition is that two adjacent stripes cannot have the same colour?

The IT department of a school is creating temporary access identifiers for students. The codes are generated by arranging $k$ distinct digits chosen from the set of odd numbers $\{1, 3, 5, 7, 9\}$, without repeating any digit within the same code.

1. a) If the server has registered exactly $60$ total possible identifiers, set up an equation using the definition of permutations (variations) without repetition where the unknown is the length $k$ of the code, and find its value with reasoned steps.
2. b) If the security rules changed and repeating digits were now permitted while maintaining the same length $k$ found in the previous section, how many total codes could be created? Compare both results by calculating how many additional identifiers are gained, and explain why repetition increases the options.

In a chess tournament, a double round-robin is played (each player faces all the others twice, alternating the colour of the pieces). If a total of $90$ games were played, set up the corresponding equation using the formula for variations (arrangements) and find how many players participated in the tournament.

Given that the number of variations with repetition of $x$ elements taken $2$ by $2$ exceeds by $12$ the number of variations without repetition of the same $x$ elements taken $2$ by $2$, set up the corresponding mathematical equation and find the value of $x$.