Exercises: Binomial coefficients

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:

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

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) $5! - 4!$

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 Spanish national basketball team.

A study states that $70\%$ of Spanish teenagers use TikTok daily. If we randomly select a group of $8$ teenagers from a secondary 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).

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

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

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

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 $n$?

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, $P(X \geq 1)$.

For a binomial distribution $X \sim B(100; 0,2)$, calculate:

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

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

Mateo is working on a task about organizing school activities and needs to calculate how many committees of 3 students can be formed from a group of 8 class representatives. In his notebook, he writes the following:

$$\binom{8}{3} = \frac{8!}{(8-3)!} = \frac{8!}{5!} = 8 \cdot 7 \cdot 6 = 336$$

His classmate Alba checks the calculation and warns him that the result is wrong.

1. a) Explain in detail the conceptual error made by Mateo. What quantity has he actually calculated and why does it not correspond to the formation of a committee without distinct roles?
2. b) Justify which factor is missing from the denominator of his formula, what role this factor plays in avoiding repetitions, and calculate the correct value of $\binom{8}{3}$.

For a study on elective subjects in Bachillerato, you need to calculate the value of the combination $\binom{25}{23}$ by hand. Your teacher suggests that you consider three possible paths before starting to compute:

• Method A: Apply the strict definition by fully expanding the factorials: $\dfrac{25!}{23! \cdot 2!}$, computing the products in the numerator and denominator first.

• Method B: Use the symmetry property $\binom{n}{r} = \binom{n}{n-r}$ to transform the combination into $\binom{25}{2}$ and simplify factors before multiplying.

• Method C: Draw Pascal's triangle row by row from row 0 to row 25, adding successive values using Pascal's rule (Stifel's identity).

1. a) Compare the three methods and justify which one is the most efficient in this specific case, and why the other two are not advisable in a time-limited written exam.
2. b) Apply the chosen method to calculate the final result, showing all simplification steps.

At a high-performance sports center, a group of 4 athletes is to be selected to represent Spain in an international relay competition. If there are 12 athletes available at the center (7 men and 5 women), calculate:

1. a) The total number of different ways the team of 4 people can be chosen.

1. b) In how many of those combinations would the team consist of exactly 2 men and 2 women?

1. c) Use the properties of binomial coefficients to prove, without calculating the numerical value, that:

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

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?

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 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).

In a vaccination campaign at a secondary 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?

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?

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 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?

A public opinion consultancy firm in Spain wants to select a consumer panel from a volunteer group of 12 people. The technical team is debating three possible sizes for the panel: 3 people, 9 people, or 6 people.

1. a) Without performing any prior numerical calculations, make a reasoned estimate of whether there will be more ways to form a 3-person panel ($\binom{12}{3}$) or a 9-person panel ($\binom{12}{9}$), or if both numbers will be equal, relying on the meaning of 'choosing who gets in' versus 'choosing who is left out'.
2. b) Check your estimate by formally calculating $\binom{12}{3}$ and $\binom{12}{9}$ using the factorial formula.
3. c) Before calculating $\binom{12}{6}$, make a reasoned estimate of whether the number of possible combinations for a 6-person panel will be greater than, less than, or equal to $\binom{12}{3}$. Next, calculate $\binom{12}{6}$ and assess whether the result matches the position that this value occupies in the corresponding row of Pascal's triangle.

At an on-call primary care health centre in Andalusia, management must organise night shifts using pairs of 2 physicians chosen from a fixed staff of $n$ available professionals. Management is considering hiring 1 more physician, which would increase the staff to $n+1$ professionals and allow for 7 more possible pairs than before.

1. a) Set up an equation relating the combinations $\binom{n+1}{2}$, $\binom{n}{2}$, and the announced increase in pairs.
2. b) Solve the equation to determine the number $n$ of physicians currently on staff.
3. c) Prove, by applying Pascal's rule (Stifel's identity) $\binom{n}{k} + \binom{n}{k-1} = \binom{n+1}{k}$, that the difference $\binom{n+1}{2} - \binom{n}{2}$ is directly equivalent to $\binom{n}{1}$, checking that your previous result is obtained immediately without the need to solve a quadratic equation.

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: $\displaystyle \binom{n}{k} = \frac{n!}{k!(n-k)!}$.

A well-known social network is analysing the behaviour 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 $4$.

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?

In the mathematics department of a secondary 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?

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).

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.

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 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).

According to data from the INE's Survey on Essential Population Characteristics, in a large Spanish metropolitan area the probability that a regular commuter train user rates the service's punctuality positively is $p = 0,3$ ($30\%$). At a station, 5 passengers are independently surveyed at random.

1. a) Define the random variable $X =$ "number of passengers who rate punctuality positively among the 5 surveyed". Justify in a reasoned manner whether it is a discrete or continuous variable and check whether it meets the requirements to be modelled using a binomial distribution $B(n, p)$.
2. b) In the formula $P(X=2) = \binom{5}{2} \cdot 0,3^2 \cdot 0,7^3$, explain in your own words what specific role the combination $\binom{5}{2}$ plays compared to the factors $0,3^2$ and $0,7^3$. Why would it be incorrect to omit the combination when calculating the probability?
3. c) A spokesperson for a commuters' association states: "Since the individual probability of satisfaction is barely $0,3$, it is almost three times as likely to find 1 satisfied user as it is to find 2 satisfied users". Model both probabilities ($P(X=1)$ and $P(X=2)$), calculate their exact quotient, and argue mathematically whether the spokesperson's statement is correct or if it makes an error by ignoring the weight of the combination compared to the individual probabilities.

The works council of a technology company in Madrid is calling an internal process to set up a negotiating committee with a number $k$ of representatives chosen from a slate of $n$ workers. Two pieces of data are available regarding the process:

1) If it were decided to form a minimum committee of only 2 representatives, there would be exactly 45 different combinations of candidates: $\binom{n}{2} = 45$.

2) For reasons of internal representativeness, the number of possible committees of size $k$ is exactly the same as the number of possible committees if the committee is expanded to $k+2$ members: $\binom{n}{k} = \binom{n}{k+2}$, where $k > 0$.

1. a) From the first piece of data, set up an equation and deduce in a reasoned manner the total number $n$ of workers on the slate.
2. b) Relying on the symmetry property of combinations ($\binom{n}{r} = \binom{n}{n-r}$), establish a linear relationship between $n$, $k$, and $k+2$, and find the exact value of $k$.
3. c) If the slate were ultimately to increase by 1 person (reaching $n+1$ members) and the committee size were set at $k+1$ members, use Pascal's rule (Stifel's identity) to calculate how many different committees could be formed by adding two combinations from the original slate, and interpret the meaning of each of those terms in the context of the company.