Exercises: Combinations

Calculate the numerical value of the following combinations applying their defining formula:

1. a) $\binom{9}{3}$
2. b) $\binom{15}{0}$
3. c) $\binom{8}{8}$
4. d) $\binom{12}{10}$

Using the symmetry property of combinations, determine the numerical value of $x$ (given that it is different from 5) that satisfies the following equality:

$\binom{17}{5} = \binom{17}{x}$

Find the exact value of the following sum without needing to calculate each combination individually, applying known properties:

$\binom{6}{0} + \binom{6}{1} + \binom{6}{2} + \binom{6}{3} + \binom{6}{4} + \binom{6}{5} + \binom{6}{6}$

Solve the following equation, determining the value of the natural number $x$:

$\binom{x}{2} = 28$

Expand the mathematical expression $(2a + b)^4$ applying the binomial theorem.

Write down the elements that make up the sixth row of Pascal's triangle, remembering that the first row corresponds to $n=0$, and use it to expand the binomial $(x - 2)^5$.

Determine only the numerical coefficient of the term in $x^3$ in the expansion of the binomial $(x + 4)^5$.

Express the sum $\binom{11}{4} + \binom{11}{5}$ as a single combination using Pascal's identity (recurrence relation) and then calculate its numerical value.

During an environmental volunteering campaign in Doñana National Park (Huelva), 8 people must be selected from a group of 10 applicants to form a trail monitoring team. Sergio, one of the participants, tries to find out how many different teams can be formed and writes on the board:

1. a) Identify and explain the conceptual error Sergio made when applying the formula. What type of combinatorial arrangement did he actually calculate by not including the term $(10-8)!$ in the denominator?
2. b) Correct the calculation by applying the symmetry property $\binom{n}{k} = \binom{n}{n-k}$ to justify why it is much more direct to calculate $\binom{10}{2}$, and obtain the actual number of possible teams.

In a 4th-year ESO class with 25 students, 2 people will be chosen to represent them on the school council. The teacher points out that the number of possible pairs can be written as $\binom{25}{2}$ or as $\binom{25}{23}$. To obtain the numerical result, three members of the group propose different approaches:

• Lucía proposes expanding the complete factorial formula: $\frac{25!}{23! \cdot 2!}$, calculating all factorials down to 1 before simplifying.

• Marcos proposes applying the symmetry property $\binom{25}{23} = \binom{25}{2}$ and simplifying the fraction $\frac{25 \cdot 24}{2 \cdot 1}$ before performing any multiplication.

• Alba proposes constructing Pascal's triangle row by row from row 0 up to row 25 to read the corresponding entry directly.

1. a) Compare the three proposals by analyzing their advantages and disadvantages. Which of the methods is the most efficient in practice, and which turns out to be completely unsuitable for a written test? Justify your answer.
2. b) Apply the most suitable method to calculate the final value and state a simple rule about when it is convenient to use the symmetry property of combinations.

In a biotechnology laboratory, 12 different bacterial strains are available. To carry out a resistance trial, a sample consisting of exactly 5 strains must be selected. Express the total number of possible samples using combination notation and calculate its value. How many of these possible samples include a specific strain that is already known to be antibiotic-resistant?

Determine all natural values of $x$ that satisfy the following combinatorial equation:

$\binom{20}{2x} = \binom{20}{x+2}$

Calculate and simplify as far as possible the fourth term in the expansion of the binomial $(3x - \frac{y}{2})^7$, ordering the terms according to descending powers of $x$.

Simplify the following algebraic expression, expressing the final result as a polynomial in terms of $n$:

$\frac{\binom{n}{4}}{\binom{n}{3}}$

In a Galton board used for probability demonstrations, a ball falls through 6 rows of pegs. At each row, the ball hits a peg and can bounce to the left or to the right. Express using a combination the number of different paths that lead to the third bin from the left (considering the leftmost bin as bin zero) and calculate this value.

A pizzeria offers a standard cheese and tomato base to which up to 8 different extra toppings can be added. A customer can choose anywhere from adding no extra toppings to adding all of them. Express the total number of different pizza combinations that can be ordered as a sum of combinations and show numerically that this value matches $2^8$.

Expand completely and simplify as far as possible the following algebraic expression using the binomial theorem:

$(x^2 - \frac{2}{x})^4$

Prove, using Pascal's identity (recurrence relation of combinations) $\binom{n}{k} + \binom{n}{k+1} = \binom{n+1}{k+1}$, that the following equality is true without needing to calculate the numerical value of each combination:

$\binom{10}{4} + 2\binom{10}{5} + \binom{10}{6} = \binom{12}{6}$

A traditional tapas restaurant in Logroño offers 8 different Riojan portions on its menu. For group dinners, they have created the promotion "Tasting your way", which allows the table to order any selection of those 8 portions: from ordering 0 portions (if they only have drinks) up to ordering all 8 portions, including combinations of 1, 2, 3, 4, 5, 6, or 7 different portions without repeating any.

1. a) Before doing detailed calculations, make a reasoned estimate: will the total number of different possible orders be closer to 50, 250, or 1000? Explain what you base your initial estimate on without adding term by term.
2. b) Apply the property of the sum of the combinations of a row of Pascal's triangle ($\binom{8}{0} + \binom{8}{1} + \dots + \binom{8}{8} = 2^8$) to find the exact number of possible orders.
3. c) Check whether your initial estimate was accurate. If the restaurant requires that each group must order at least one portion and decides to add 2 new tapas to the menu (having 10 in total), does the number of available options double or quadruple compared to the value obtained in part b? Justify your deduction using the properties of powers.

A bicycle courier delivers packages in the Eixample district of Barcelona, whose streets form a regular grid. The cyclist must leave the courier hub (point $A$) and reach the delivery address (point $B$), located 4 streets east and 3 streets north of $A$. To optimize battery life, they only move east ($E$) or north ($N$) along paths of minimum length (7 street blocks in total).

1. a) Express the total number of different minimum routes the courier can follow using a combination of the form $\binom{m+p}{m}$ and calculate its numerical value.
2. b) Due to construction work to remodel a bike lane, intersection $C$, located 2 streets east and 1 street north of $A$, is completely closed. Calculate how many minimum routes go from $A$ to $C$ and how many go from $C$ to the final destination $B$.
3. c) Using the previous results, deduce how many minimum routes are left for the cyclist to go from $A$ to $B$ without passing through point $C$ under construction. Explain the reasoning that allows you to solve it by subtraction.

Find the independent term (that is, the term that does not depend on the variable $x$) in the expansion of the binomial $(2x^2 - \frac{1}{x})^9$.

Prove algebraically, starting from the factorial definition $\binom{n}{k} = \frac{n!}{k!(n-k)!}$, the validity of Pascal's triangle recurrence relation:

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

Solve the following combinatorial equation, determining the value of the natural number $x$:

$\binom{x+1}{2} + \binom{x}{2} = 36$

Calculate the exact value of the expression $(\sqrt{5} + 2)^6 + (\sqrt{5} - 2)^6$. Apply the expansion using the binomial theorem and observe how the terms containing roots with an odd index cancel each other out.

A robot in a logistics warehouse moves through a grid network of aisles from the starting point $A(0,0)$ to the destination point $B(6,4)$, moving only one step to the right or one step upwards at each iteration. Due to maintenance work, the robot cannot pass through node $C(3,2)$. Express the number of valid paths using combinations and calculate the final result.

In the expansion of the binomial $(a + b)^n$, it is known that the numerical coefficients of the fourth and fifth terms are exactly equal. Determine the value of the natural exponent $n$.

Use the expansion of the binomial theorem evaluated at the expression $(1 - 1)^n$ to prove that the alternating sum of the elements in any row of Pascal's triangle (for $n > 0$) is always equal to zero:

$\binom{n}{0} - \binom{n}{1} + \binom{n}{2} - \dots + (-1)^n \binom{n}{n} = 0$

A technology company needs to form a 5-member cybersecurity team, selected completely at random from a staff of 8 programmers and 6 systems analysts. Calculate the probability that the resulting team consists of exactly 3 programmers and 2 analysts. Express both the favourable outcomes and the possible outcomes using combinations before giving the resulting irreducible fraction.

In a rural area of Castilla y León, a school volleyball tournament is organized during the first term. The sports committee establishes that in the preliminary phase, all registered schools must play exactly one match against each of the other teams (single round-robin system). At the end of the regular phase, the federation certifies that exactly 45 matches were played in total.

1. a) Model the situation by setting up an equation involving a combination of the form $\binom{n}{2}$ set equal to the total number of matches, clearly defining what the unknown $n$ represents.
2. b) Apply the formula for combinations to transform your expression into a quadratic equation in standard form ($a \cdot n^2 + b \cdot n + c = 0$). Solve it and interpret the result in the context of the problem, justifying why you must discard one of the two mathematical solutions.
3. c) If next year the sports campaign is successful and twice as many schools register ($2n$), will exactly twice as many matches be played? Prove your answer by calculating how many matches would be organized with the new number of participants, and reflect on the growth rate of combinations compared to linear growth.

In a public hospital in the Community of Madrid, an on-call medical committee consisting of 5 professionals must be formed. To select them, an on-call team consisting of 7 female doctors and 5 male nurses (12 healthcare workers in total) is available.

1. a) The hospital management establishes that the committee cannot be exclusive to a single professional category: it must have at least one female doctor and at least one male nurse. Explain in detail why it is much quicker to calculate the total number of possible committees and subtract the non-permitted cases (complementary method) than to add the valid options case by case. Calculate the number of permitted committees using this method.
2. b) One of the 7 female doctors is the general coordinator of the service. She notes that, when forming the 5-person committee, there are only two mutually exclusive situations regarding her: either she is necessarily part of the committee, or she is excluded to supervise external emergencies.

Relate this fact to Pascal's rule:

Explain what each of the three combinations in this equality represents in the hospital context and justify, in your own words, why the sum of the two terms on the right matches the value on the left exactly, with no possibilities missing or duplicated.

Topic challenge. Is it easier to win the lottery or have your school password guessed?

1. Research: Check the rules of La Primitiva (choosing 6 numbers out of 49) on the Loterías y Apuestas del Estado (SELAE) website and basic guidelines for creating strong passwords on the INCIBE website.

2. Combinatorial calculation: In a spreadsheet, determine:

3. The total number of possible bets using the binomial coefficient $\binom{49}{6} = \frac{49!}{6! \cdot 43!}$, applying factorials.

4. The resulting permutations if the order of ball extraction mattered.

5. The permutations of 4 distinct digits for a locker code compared to letter combinations.

6. Report and proposal: Write a one-page report comparing the mathematical difficulty of winning the draw (with a probability of barely $0,0000071\,\%$) against the vulnerability of a simple PIN. Put forward a reasoned proposal for a secure password using your calculations.

Assessment criteria:

• Accuracy in calculating factorials, permutations, and binomial coefficients in the spreadsheet.

• Proper collection and application of data from official sources (SELAE and INCIBE).

• Consistency and mathematical rigour in the conclusions and recommendations of the final report.