Counting Techniques: Permutations and Combinations

To celebrate the end of the school year, a group of friends decides to go for dinner at a restaurant that offers a set menu. The students must choose an appetizer from 4 possible ones, a main course from 5 options, and a dessert from a list of 3. In how many different ways can a student configure their dinner menu?

A high school student wants to create a secure password for their new profile on a social network. The password must have exactly 4 numerical digits (from $0$ to $9$). If numbers are allowed to be repeated, how many different code combinations could they generate?

A new online banking service requires its users to create a personalized security code. The code must consist of exactly 4 digits followed by 2 letters (using the 26 letters of the English alphabet).

Calculate how many different codes can be generated in the following scenarios:

1. a) The digits can be repeated, but the letters must be different.
2. b) Both the digits and the letters can be repeated, but the first digit cannot be zero and the last letter must be a vowel ($A, E, I, O, U$).

In the school library, a student has selected 5 different literature books and wants to place them on a shelf in their room. In how many different ways can they order them from left to right?

How many 5-letter words, with or without meaning, can be formed by permuting the letters of the word $LIBRO$? And what if all the words had to start with the letter $L$?

For the group stage of an academic debate tournament, the speeches of 8 different schools must be organized. The organizing team has decided that the opening debate will always be held by Athena Academy and the closing debate will be concluded by Newton High School.

In how many different ways can the order of speeches of the remaining 6 schools be configured to complete the tournament program?

8 players participate in an esports competition. If only the top three classified players are awarded gold, silver, and bronze medals, respectively, how many different podiums can be formed at the end of the tournament? Bear in mind that the order in which the players finish does determine the type of medal.

Solve the following combinatorial equation with reasoning to find the value of $n$:

Knowing that $V_{n, k}$ represents the permutations of $n$ elements taken $k$ at a time, $k$.

On a shelf in a home decor shop, they want to place in a row 9 vases identical in shape but of different colors: 4 white vases, 3 blue vases, and 2 silver vases.

In how many different ways can these 9 vases be lined up on the shelf?

Eight friends, including a couple, go to dinner at a Japanese restaurant and are assigned a circular table.

1. In how many ways can they sit at the table if the couple wants to sit together?

2. If after dinner they have a falling out and decide that at the movie theater (where they have 8 seats in a row) they do not want to sit together nor have anyone between them (that is, there must be at least one empty seat or another person separating them, but never adjacent seats), in how many ways can the eight friends sit in the row of seats?

A digital marketing team is designing an advertising campaign for a well-known sneaker brand. For a photo shoot, they must place their 10 star collaborators in a line: 3 photographers, 4 video editors, and 3 creative writers (copywriters). Determine in how many ways they can pose for the photo if the following conditions must be met simultaneously:

1. Professionals in the same category must always be together.

2. The main photographer and the chief writer cannot be at the ends of the total formation.

An ice cream parlor offers 12 different ice cream flavors. If we want to buy a tub with 2 scoops of different flavors, how many possible ice cream combinations can we choose? Consider that the order in which the scoops are placed in the tub does not change the final product.

A basketball coach has 10 players on the roster for a game. If for the tip-off they must choose a starting lineup of 5 players and the specific position each one occupies at that moment does not matter, how many different starting teams can they present?

A senior class has to choose a student committee made up of 3 students. If there are 25 students in the class and all are eligible, how many different groups of 3 people can be formed? Reason whether or not the order of selection of the members matters in this case.

A chef is designing a tasting menu for a gala dinner. The menu must consist of 3 appetizers, 2 main courses, and 2 desserts.

If in their recipe book they have 6 appetizer options, 4 main course options, and 3 dessert options, how many different menu combinations can they offer their customers?

In a video game development company, the art department has a staff of 12 designers, of which 7 are experts in 3D modeling and 5 are experts in texture design.

For a new project, a work team consisting of 4 designers must be chosen. In how many ways can the team be chosen if it is required that there are at least two experts in 3D modeling?

A quick pickleball tournament is to be organized with 12 students from a class. The teacher wants to form 6 different pairs.

1. In how many ways can the 6 pairs be formed if the order of the pairs and the order of the players within each pair do not matter?

2. If among the students there are 4 who are advanced level players and the teacher decides that each of them must be in a different pair (to balance the level), in how many ways can the pairs be formed now?

A blitz chess tournament is going to be held in a youth center. At the end of the qualifying phase, there are 10 players left. The organization must award three different trophies: one to the champion, another to the runner-up, and a third to the third-place finisher. In addition, a participation medal will be awarded to the remaining 7 players.

1. a) In how many ways can the three main trophies be distributed?
2. b) If among the 10 finalists there is a pair of siblings and we want to know in how many of those classifications the two siblings receive a trophy (regardless of which one it is).

To access a VIP area of a music festival, attendees must generate a 7-character personal code. The organization's rules are strict to prevent hacking:

• The first 3 characters must be different capital letters (from a 26-letter alphabet).

• The remaining 4 characters must be digits from $0$ to $9$, which can be repeated.

• Additional restriction: The code is not valid if the three central digits of the numerical block are equal to each other (for example, ABC-1222 or ABC-5550 would not be valid, but ABC-5505 would be).

Calculate how many different codes can be created under these conditions.

14 players participate in a school esports league. A committee of 5 representatives must be selected for the national federation. Among the 14 players, there is a pair of siblings who do not get along at all and have declared that they will not be part of the committee if the other is also chosen. Furthermore, there must be at least one representative of the 3 most experienced players on the committee. In how many different ways can the 5-person committee be chosen?

An ice cream parlor offers its customers the possibility of creating a "Special Cup" by choosing 3 scoops of ice cream from its 15 available flavors.

1. a) If a customer wants the 3 scoops to be of different flavors, how many different cups can they create?
2. b) If the customer is an enthusiast of a particular flavor and allows the scoops to be of the same flavor (that is, they can repeat flavors), how many cup combinations are possible now?

Three friends have won a raffle for 12 identical tickets to a theme park and decide to share them among themselves.

1. In how many ways can the tickets be shared if each must receive at least one?

2. And what if they decide that the distribution should be such that one of the friends (the one who bought the raffle ticket) always receives more tickets than the other two combined?

A messaging application uses an encryption system based on 12-bit binary sequences (zeros and ones). For reasons of algorithm efficiency, each sequence is required to meet two properties:

• It must contain exactly six zeros and six ones.

• Critical condition: There cannot be three or more consecutive ones anywhere in the sequence.

Calculate the number of valid sequences that this system can generate.

Consider the binomial expansion for the expression:

Without expanding the whole power, calculate:

1. The value of the coefficient of the term containing $x^{20}$.

2. Determine with reasoning if there is any term independent of $x$ in this expansion. If so, indicate its position.