Combinatorial Numbers

Calculate and simplify the value of the following expressions without using a calculator, applying the properties of factorials:

1. a) $\frac{8!}{6!}$

1. b) $\binom{7}{2}$

1. c) $5! - 4!$

To organize the graduation party for 2.º de Bachillerato, a party committee made up of $3$ students must be chosen. If there are $20$ volunteer students in the class, in how many different ways can this group be chosen? (The order in which the students are chosen does not matter: only who is on the committee counts.) Use the combinatorial number formula:

$$\binom{n}{r} = \frac{n!}{r!(n-r)!}$$

In the section on binomial coefficients, there is a symmetry property stating that $\binom{n}{r} = \binom{n}{n-r}$.

Verify this property by calculating the values of $\binom{10}{8}$ and $\binom{10}{2}$. What conclusion do you reach regarding the computational effort when $r$ is a number close to $n$?

In a high-performance sports center, a team of 4 athletes is to be chosen to represent Mexico in an international relay competition. If there are 12 athletes available in the center (7 men and 5 women), calculate:

1. a) The total number of different ways the 4-person team can be chosen.
2. b) In how many of those combinations would the team consist of exactly 2 men and 2 women?
3. c) Use the properties of binomial coefficients to prove, without calculating the numerical value, that:

$$\binom{12}{4} = \binom{11}{3} + \binom{11}{4}$$

Prove, using the definition of binomial coefficient using factorials, the following identity (known as Pascal's rule or Stifel's identity):

$$\binom{n}{k} + \binom{n}{k-1} = \binom{n+1}{k}$$

Apply this result to calculate the value of $\displaystyle \binom{10}{4} + \binom{10}{3}$ without calculating each term separately.

Solve the following questions related to combinatorics and factorials:

1. a) Find the value of $n$ in the following equality:

$$\binom{n}{2} = 21$$

1. b) Simplify the following expression, leaving the result in terms of $n$:

$$\frac{(n+1)! + n!}{(n-1)!}$$

1. c) How many 5-digit numbers can be formed using only the digits $\{1, 2, 3, 4, 5\}$ if no digit may be repeated? How would the result change if repetitions were allowed?

In the math department of a high school, they want to form a graduation committee. There are $12$ Science students and $10$ Social Sciences students.

1. a) In how many ways can a committee of $6$ people be chosen if there must be at least $4$ Social Sciences students? Use the properties of binomial coefficients to simplify manual calculations.

1. b) If the selection process for each member were independent and the probability of choosing a Social Sciences student were the actual proportion in the group ($10/22$), what would be the probability that, in a committee of $6$ people chosen with replacement, there were exactly $3$ from each pathway?

Before studying the binomial distribution, it is essential to distinguish the type of variable. Classify the following random variables as discrete or continuous:

1. The number of "likes" an Instagram post receives in one hour.

2. The exact time it takes a student to get to school.

3. The number of people who test positive in an antigen test in a sample of $50$ individuals.

4. The height of the players in the Mexican national basketball team.

Given that for a binomial distribution $B(n, p)$, the probability of obtaining $k$ successes is:

$$P(X = k) = \binom{n}{k} \times p^k \times (1-p)^{n-k}$$

Calculate the probability of obtaining exactly $2$ heads when tossing a fair coin $4$ times.

A study states that $70\%$ of Mexican teenagers use TikTok daily. If we randomly select a group of $8$ teenagers from a high school:

1. a) Identify the parameters $n$ and $p$ of the binomial distribution that this situation follows.
2. b) Define the random variable $X$ associated with this experiment.
3. c) What is the probability that exactly $8$ teenagers use the app? (State only the formulation of the formula).

A young entrepreneur has launched a sustainable fashion app. According to her market research, $35\%$ of users who download the app end up making a purchase in the first week. If 10 new independent users registered today:

1. a) Define the random variable $X$ and justify whether it follows a binomial distribution, stating its parameters.
2. b) Calculate the probability that exactly 4 of these users make a purchase.
3. c) What is the probability that at most 2 users buy in the first week?

For a binomial distribution $X \sim B(100; 0.2)$, calculate:

1. a) The mean or expected value ($\mu = n \times p$).
2. b) The variance ($\sigma^2 = n \times p \times q$).
3. c) The standard deviation ($\sigma = \sqrt{n \times p \times q}$).

Briefly explain what the value of the mean represents in this context.

A multiple-choice exam consists of 12 questions, each with 4 answer choices of which only one is correct. A student who has not studied at all decides to answer all questions at random.

1. a) Calculate the expected number of correct answers (mean) and the standard deviation of this distribution.
2. b) What is the probability that the student passes the exam (obtains at least 6 correct answers)?
3. c) If the exam had 100 questions, what parameters would the resulting binomial distribution have? (Do not carry out the probability calculations).

A well-known social network is analysing the behavior of its "stories". It has been verified that the distribution of the number of users who interact with an advertisement follows a binomial model. In a sample of users, it is known that the mean number of interactions is $\mu = 12$ and the variance is $\sigma^2 = 7.2$.

1. a) Calculate the sample size $n$ and the interaction probability $p$ for each user.

1. b) Once $n$ and $p$ are obtained, determine the probability that, in a new sample of the same size, the number of interactions is greater than $2$ but less than or equal to $4$.

In a vaccination campaign at a high school, it has been observed that $10\%$ of the students experience a mild side effect (such as arm pain). If a class of 25 vaccinated students is selected:

1. a) What is the expected mean number of students to have side effects?
2. b) Calculate the probability that the number of students with side effects is less than the mean.
3. c) What is the probability that exactly 3 students have side effects?

A study on reading habits in 17-year-old teenagers states that $70\%$ read regularly in digital format. $10$ teenagers are randomly selected.

1. a) Calculate the probability that the number of teenagers who read digitally equals the mode of this binomial distribution. (Note: The mode in a $B(n, p)$ is the integer value $k$ such that $(n+1)p - 1 \leq k \leq (n+1)p$).

1. b) If it is known that at least $8$ of them read in digital format, what is the probability that all $10$ do? (Conditional probability).

A multiple-choice exam consists of $5$ true-or-false questions, and the answers to some questions do not affect the answers to others (they are independent events). A student answers all of them at random, so that on each question the probability of getting it right is $p=0.5$. Let $X$ be the number of questions the student gets right.

1. a) What is the probability that the student gets no question right, that is, $P(X=0)$?

1. b) Using the complementary event, explain how you would calculate the probability that the student gets at least one question right, $P(X \geq 1)$.

It is known that $15\%$ of the containers produced by a plastic recycling plant do not meet the required purity standards. A quality inspector randomly selects 8 containers for analysis. Assume that the containers are chosen independently and that the probability that each one is defective is always $0.15$.

1. a) Calculate the probability that all the selected containers are acceptable (meet the standards).
2. b) Calculate the probability that at least one of the containers is defective.
3. c) What is the probability that there are more than 6 acceptable containers in the sample?

A TikTok content creator knows that the probability of one of his videos going "viral" (exceeding 100 000 views) is 0,05. If he decides to post one video a day for a week (7 days), and we assume the videos are independent of each other:

1. a) What is the probability that none of his videos goes viral?

1. b) Calculate the probability that at least 2 videos are viral.

1. c) Draw a tree diagram showing the possibilities of success (V, viral video) and failure (N, non-viral video) for the first 3 days and assign the probabilities to each branch.

You can do it on paper and send a photo with your answer.

A delivery app has a probability of delivery delay of $p = 0.15$. A courier makes $20$ deliveries during a night shift. Let $X$ be the number of delayed deliveries (deliveries are assumed to be independent, so that $X$ follows a binomial distribution $B(20;\ 0.15)$). The figure shows the probability distribution of $X$.

1. a) What is the probability that there are exactly $3$ delays? Express the binomial coefficient as a product of fractions before calculating.

1. b) If the courier receives a penalty for having more than $4$ delays, what is the probability of being penalized? (Set up the calculation using the complementary event and justify why it is more efficient in this case, comparing how many individual probabilities would have to be calculated with each method).

A resit exam consists of $80$ multiple-choice questions with $4$ options each, only one of which is correct. A student randomly guesses the $12$ questions they did not know.

1. a) Let $X$ be the variable "number of correct answers in the $12$ questions". Define the parameters of the distribution.

1. b) Calculate the probability that they have more correct answers than incorrect ones.

1. c) If passing that section requires at least one correct answer, calculate the probability that they fail.

A smartphone component manufacturing company carries out quality control. Each production batch of a critical part contains $2\%$ defective units. The parts are packed in boxes of $15$ units. A customer buys a shipment of $100$ boxes.

1. a) What is the probability that a randomly chosen box contains at least $2$ defective parts?

1. b) If the customer returns the box if it contains more than one defective part, how many boxes are expected to be returned out of the $100$ bought?

A courier company uses drones to deliver packages in rural areas. It has been verified that the probability of a drone making a landing accuracy error is 0.12. If a drone performs 15 landings in a day:

1. a) Calculate the probability that it succeeds in all landings.
2. b) Calculate the probability that it makes errors in more than 2 landings.
3. c) If the company wants the mean number of errors per day to be at most 1, how many landings should the drone perform per day based on the same error rate?

In an e-sports tournament, the probability of a professional "League of Legends" player achieving a "Pentakill" in a competitive match is $p = 0.008$. A team decides to participate in a league consisting of a certain number of matches $n$. It is known that the probability of the team not getting any "Pentakills" in the entire league is less than $0.6$.

1. a) Determine, using logarithms, the minimum number of matches the team must play for this condition to be met.

1. b) If they end up playing $120$ matches, calculate the probability that they achieve exactly two "Pentakills", expressing the final result to four decimal places. To calculate the binomial coefficient, use the factorial formula: $\displaystyle \binom{n}{k} = \frac{n!}{k!(n-k)!}$.