Exercises: Ways of counting

In a school cafeteria, the menu of the day allows choosing between 3 appetizers, 4 main courses, and 2 desserts. In how many different ways can a student configure their complete lunch consisting of an appetizer, a main course, and a dessert?

A group of 10 friends decides to organize a video game tournament. Prizes will be awarded to the first, second, and third place finishers. Knowing that there can be no ties, in how many different ways can these three trophies be distributed?

To log into a mobile app, a user must create a 4-digit security code. If the system allows numbers to be repeated and uses the digits from $0$ to $9$, how many different security codes can be generated?

A bookshelf has space to place exactly 6 textbooks from different senior year subjects. In how many different ways can these 6 books be arranged on the shelf if we want them all to be aligned next to each other?

In a class of 12 students, the homeroom teacher must choose 3 of them to be part of the school's student council. Given that the three chosen will have the same responsibility, how many different groups of representatives can be formed?

How many 4-letter words, whether they make sense or not, can be formed using the letters of the word BALA? (Keep in mind that the letter 'A' is repeated).

An ice cream parlor offers 8 different flavors of ice cream. If a customer wants to buy a cup with 2 scoops of ice cream of different flavors, how many different combinations can they choose?

A social network requires usernames to always start with a lowercase letter (from the 26 in the English alphabet) followed by 2 digits (from $0$ to $9$). How many different usernames under this format can be created?

To improve the security of the smart lockers in a school, the system requires an access code of $4$ digits. However, to avoid obvious combinations, the system imposes the following conditions: the first digit must be odd, the last must be even (considering $0$ as even), and no digit can be repeated in the code. How many different codes can be generated following these rules?

$12$ teams participate in a regional esports competition. The organizers must award a gold, a silver, and a bronze medal to the top three finishers.

1. a) In how many different ways can the podium of the competition be formed?
2. b) If we know that the local team "The Gamers" always finishes in the top three, in how many ways can the podium be completed now?

An urban clothing brand wants to launch a limited edition of hoodies. They have $3$ identical red hoodies, $2$ identical blue hoodies, and $4$ identical black hoodies. For a photo shoot, they must place all these hoodies in a row on a counter. In how many different ways can the hoodies be arranged based solely on their color?

A group of $15$ students from $2º$ grade is organizing the graduation party. They must choose a committee made up of $4$ people to be in charge of the music and food.

1. a) How many different committees can be formed?
2. b) If there are $8$ girls and $7$ boys in the group, how many of those committees would be made up of $2$ girls and $2$ boys?

Seven friends go to dinner at a pizzeria and sit at a circular table.

1. a) In how many different ways can they sit around the table?
2. b) Two of the friends, Ashley and Ethan, have argued and do not want to sit together. In how many ways can the seven friends sit now fulfilling this condition?

In a cafeteria they offer a "Student Combo" which consists of choosing a type of sandwich, a side dish, and a drink. They have $5$ types of sandwiches, $3$ types of side dishes, and $4$ types of drinks.

1. a) How many different meals can be configured?
2. b) If a student decides that they do not want a veggie sandwich (there is one type) nor do they want to drink water (there is one type), how many options do they have now for their meal?

A courier must deliver a package in a city whose street structure is a perfect grid. They must go from point $A$ (bottom-left corner) to point $B$ (top-right corner), moving only to the right or upward. If the distance is $6$ blocks to the right and $4$ blocks upward:

1. a) How many different routes can they follow?
2. b) How many of those routes pass through a checkpoint $C$ located $3$ blocks to the right and $2$ upward from point $A$?

Find the value of $n$ in the following equation with binomial coefficients, reasoning the steps taken:

In the binomial expansion for the expression $(\frac{x^2}{2} - \frac{k}{x^3})^{15}$, it is known that the coefficient of the term containing $x^{10}$ is equal to $3003$. Determine with reasoning the value of the positive real number $k$ and, once found, calculate the value of the constant term of this expansion if it exists.

Determine all the values of $n \in \mathbb{N}$ that satisfy the following algebraic equality based on binomial coefficients:

Justify the process by expanding the factorials and solving the resulting equation.

A cybersecurity company wants to generate unique access codes for its employees. Each code must have a length of 9 characters distributed as follows:

• 3 initial capital letters (chosen from the 26 in the alphabet).

• 4 central numerical digits (0-9).

• 2 final special symbols (chosen from a set of 5 symbols: @, #, $\$$, &, *).

Calculate how many different codes can be generated if the following restrictions are imposed:

1. The capital letters cannot be repeated.

2. The numerical digits must all be different and arranged in strictly increasing order.

3. The special symbols can be repeated.

An architecture studio is designing a housing development with 7 identical lots in a row. They must plant trees of three different species: oaks, pines, and holm oaks. For aesthetic and biological reasons, the following requirements must be met:

• Only one tree is planted in each lot.

• There must be at least one tree of each species.

• There cannot be two pines in adjacent lots.

Calculate the total number of different ways in which the trees can be distributed in the 7 lots.

A programmer is developing an algorithm to find the shortest route in a sensor network represented by a grid of $6 \times 5$ nodes. The algorithm must calculate how many paths of minimum length exist from node $A(0,0)$ to node $B(6,5)$, taking into account that due to a fault, the central node $C(3,3)$ is inaccessible and cannot be part of any route.

A debate committee for a Model United Nations (MUN) consisting of 6 students is to be formed. The available candidates are 8 seniors and 6 juniors.

Calculate in how many ways the committee can be chosen if these two conditions must be met simultaneously:

1. There must be more seniors than juniors on the committee.

2. There must be at least one student from each grade in the selected group.

Six friends (3 boys and 3 girls) sit down to dinner at a circular table to celebrate the end of their exams. Determine the number of possible arrangements in which they can sit if:

1. There are no gender restrictions.

2. The boys and girls must sit alternately (no boy next to another boy and no girl next to another girl).

3. A specific pair of friends, Emma and Michael, insist on sitting together.

(Note: Consider that in circular permutations, only the relative position of one person to another matters).

A streaming platform uses an algorithm to recommend playlists. If they have 10 rock, 8 pop, and 6 indie songs available, how many 5-song playlists can be created so that each list contains at least one song from each of the three musical genres? Keep in mind that the order of the songs in the list matters.