Exercises: Without ordering

In a chess tournament, $14$ players participate. If each player must play against all the others exactly once, determine the total number of matches that will be played in the tournament.

To make a salad, you have $8$ different ingredients available (lettuce, tomato, carrot, onion, cucumber, tuna, sweetcorn and olives). Calculate how many different salads of $4$ ingredients you can make.

An exam consists of $10$ questions. The teacher specifies that students must choose and answer only $6$ of them. Determine in how many different ways a student can select the questions to answer.

In a business meeting, there are $12$ people. At the end, everyone says goodbye by shaking hands once with each other. Calculate how many handshakes take place in total.

Find the value of $x$ in the following equality of combinations, knowing that $x \neq 4$:

$\binom{x}{4} = \binom{x}{7}$

Calculate the value of the following expression using the properties of combinations:

$\binom{8}{3} + \binom{8}{4}$

From a deck of $40$ cards, $3$ cards are drawn simultaneously at random. Determine how many different outcomes can be obtained.

$7$ points are marked on a circle. Calculate how many different line segments can be drawn joining two of these points.

For the 4th Year English course, the teachers provide a list of 7 recommended readings, from which each student must choose 3 to complete an assignment during the term. Seán states with confidence: "To choose the first book I have 7 options; for the second, 6; and for the third, 5. Therefore, altogether there are $7 \times 6 \times 5 = 210$ possible ways to choose my 3 readings."

1. a) Identify the conceptual error made by Seán and explain why his calculation produces a number that is much larger than the actual number of choices.
2. b) How many times is Seán counting each possible group of 3 books? Relate your answer to the permutations of those 3 elements.
3. c) Write the correct formula using the combination notation $\binom{n}{k}$ and calculate how many distinct selections of 3 books can actually be made.

In a 4th Year class of 24 students, a student council delegation of 2 representatives must be chosen to attend the school's meetings.

1. a) Before applying any formula, make a reasoned estimate without doing meticulous calculations: do you think the number of possible pairs will be in the tens, hundreds, or thousands? Support your answer by thinking about the options each student has.
2. b) Model the situation using the appropriate combination formula, carry out the calculation, and check whether your initial estimate was accurate.
3. c) Based on the symmetry property of combinations $\binom{n}{k} = \binom{n}{n-k}$, explain why choosing the 2 representatives of the delegation is mathematically equivalent to choosing the 22 students who are left out, without needing to calculate additional factorials.

A company needs to form a project team consisting of $3$ engineers and $2$ economists. If there are $8$ engineers and $5$ economists available on staff, calculate how many different teams can be formed.

A bakery sells $6$ different types of cakes. Determine in how many different ways $4$ cakes can be bought, keeping in mind that several cakes of the same type can be chosen.

Calculate the total number of diagonals that can be drawn in a regular polygon with $10$ sides (decagon).

Solve the following equation to find the value of the natural number $n$:

$C_{n, 2} = 45$

$5$ cards are drawn simultaneously from a deck of $40$ cards (divided into $4$ suits of $10$ cards each). Determine in how many of the possible outcomes there are exactly $3$ cards of the suit of diamonds.

A robot moves along a grid from point $A$ to point $B$. It can only move right or up, advancing one intersection per step. Calculate the number of different paths it can follow.

You have $8$ identical concert tickets and you want to distribute them among your $4$ best friends. Calculate in how many ways you can distribute them, knowing that a friend may receive several tickets or even none.

In a questionnaire with $12$ questions, you must answer a total of $8$ questions. However, the first $3$ questions are compulsory. Determine in how many different ways you can complete your choice of questions.

In a regional federated padel tournament in Castilla-La Mancha, the preliminary stage is played in independent groups. In each group, all registered pairs play exactly one match against every other pair in their group (single round-robin system). When closing the preliminary stage scoresheet, the organizers confirm that exactly 45 matches were played in total in Group A.

1. a) Write a combinatorial equation using the notation $\binom{n}{k}$ involving the unknown number $n$ of registered pairs in Group A.
2. b) Algebraically expand the combination to transform the equality into a quadratic equation in standard form.
3. c) Solve the equation, analyze the two solutions obtained, reasonably discarding the one that makes no sense in this context, and state how many pairs were competing in that group.

Met Éireann, the Irish Meteorological Service, is going to audit the maintenance of an automated weather station network in Munster. It has 9 stations located in mountain areas and 6 stations in coastal areas. The technical protocol establishes that a sample of 6 stations must be selected, necessarily composed of 4 mountain stations and 2 coastal stations.

1. a) Mathematically justify why the selection for each category must be modeled using combinations without repetition and not using variations (arrangements) or permutations.
2. b) Apply the combinatorial rule of product to calculate how many different samples of 6 stations can be configured while meeting the established quota.
3. c) If, for technical reasons, it is decided that the mountain station located on Carrauntoohil must be included in the sample, calculate how many different samples of 6 stations (4 mountain and 2 coastal) can now be selected.

A hospital needs to form a night-shift team of $5$ people, selected from a group of $7$ doctors and $6$ nurses. Calculate how many different teams can be formed if regulations require that the team contains at least $3$ doctors.

In a grid consisting of $9$ points distributed in $3$ rows and $3$ columns, determine how many different triangles can be formed using these points as vertices. Bear in mind that three collinear points do not form a triangle.

Find the value of $x$ that satisfies the following combinatorial equation:

$\binom{x}{2} + \binom{x-1}{2} = 25$

The Primitiva lottery game consists of choosing $6$ numbers out of a total of $49$. Given the winning combination of $6$ numbers, calculate how many of all possible different bets would get exactly $4$ correct numbers.

A robot moves along a grid from node $A$ to node $B$, moving only to the right and upwards. It must pass through node $C$ to collect a package. Determine how many valid routes exist.

An ice cream shop offers $5$ different flavours. You want to buy $8$ tubs of ice cream for a party, but you want to make sure you take at least one tub of each of the $5$ flavours. Calculate in how many different ways you can place your order.

An urn contains $6$ red balls, $5$ blue balls and $4$ green balls. $4$ balls are drawn simultaneously. Calculate how many different outcomes contain exactly $2$ red balls and at least $1$ blue ball.

The traditional game of dominoes goes up to double six (the scores range from $0$ to $6$). A manufacturer decides to create a "super dominoes" set that goes up to double nine (the scores range from $0$ to $9$). Calculate how many tiles in total this new set will have and how many of them will not be doubles.

A medical examination board for evaluating the MIR system tests in the Ministry of Health must consist of 6 people, selected from a team composed of 4 Family and Community Medicine specialists and 7 Internal Medicine specialists. Regulations require that the resulting board must include at least one specialist in Family and Community Medicine.

Two committee members propose different methods to calculate the number of valid boards:

• Method 1 (Direct): Divide the problem into disjoint cases according to the number of family doctors (1, 2, 3, or 4), calculate the combinations for each case multiplying by the required internists, and add all results.

• Method 2 (Complementary): Calculate all possible boards of 6 people that can be formed from the 11 available doctors without any restriction, and subtract the invalid boards (those with no Family Medicine specialist).

1. a) Fully develop both methods setting up their combinations, and verify numerically that both lead to the same result.
2. b) Critically compare both methods: which do you consider more time-efficient and with less risk of calculation error in an exam situation? Justify your choice by analyzing the number of operations in each.
3. c) If the regulations changed to require that the board have "at least 3 specialists in Family Medicine", would the complementary method still be shorter than the direct one? Justify your answer by reasoning about the number of favorable cases compared to unfavorable ones.

In the orthogonal grid of streets in Georgian Dublin (around Merrion Square), a bicycle courier needs to travel from intersection A to intersection B. To optimize their route, the courier only moves east or north at each intersection, requiring an exact path of 5 blocks east and 3 blocks north (a total of 8 blocks of travel).

1. a) Model the path using combinatorics: explain why any optimal route of 8 blocks is equivalent to choosing the steps taken in one of the two directions, and calculate how many different routes connect A to B using the expression $\binom{m+p}{m}$.
2. b) The Gardaí have closed the intermediate intersection C for construction, located exactly 2 blocks east and 1 block north of point A. Model how many of the possible routes would necessarily pass through intersection C and deduce how many alternative construction-free routes remain available to the courier.
3. c) Suppose the courier must deliver an urgent package at intersection D (located 4 blocks east and 2 north of A) and that intersection C remains closed. Set up and calculate the combinatorial expression determining how many different routes allow going from A to B necessarily passing through D while avoiding the closed intersection C at all times.

Topic challenge. How do order and selection influence forming a youth committee? Design a mathematical proposal based on real demographic data.

1. Data collection: Go to the CSO (Central Statistics Office) website and find the $8$ most frequent names in your birth year (if the data are only given nationally, use the national list; otherwise use your county).

2. Combinatorial modeling: With those $8$ names as candidates, calculate and justify:

3. Election of a president, secretary, and committee member (distinct positions where order matters): variations $V_{8,3}$.

4. Coexistence committee of $3$ equal members (grouping without ordering): combinations $C_{8,3}$.

5. Possible positions for a single-file photo of all $8$ members: permutations $P_8$.

6. Spreadsheet and report: Design a spreadsheet table that automatically calculates $V_{n,3}$, $C_{n,3}$, and $P_n$ for values of $n$ between $5$ and $12$. Write a short report reasoning why $C_{n,3}$ is always smaller than $V_{n,3}$, and advise your school on which system (distinct positions or a committee without positions) is more suitable, explaining why a larger number of possible configurations does not by itself imply greater diversity in the committee.

The following will be assessed:

• Rigor in data collection from the CSO and correctness of the combinatorial calculation ($V$, $P$, and $C$).

• Effective use of the spreadsheet in automating formulas.

• Clear reasoning regarding the role of order in decision-making.